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ELEMENTARY

COMMUTATIVE ALGEBRA

LECTURE NOTES

H.A. NIELSEN

DEPARTMENT OF MATHEMATICAL SCIENCES

UNIVERSITY OF AARHUS

2005

Elementary **Commutative** Algebra

H.A. Nielsen

Contents

Prerequisites 7

1. A dictionary on rings and ideals 9

1.1. Rings 9

1.2. Ideals 11

1.3. Prime ideals 13

1.4. Chinese remainders 14

1.5. Unique factorization 15

1.6. Polynomials 16

1.7. Roots 18

1.8. Fields 19

1.9. Power series 20

2. Modules 21

2.1. Modules and homomorphisms 21

2.2. Submodules and factor modules 23

2.3. Kernel and cokernel 25

2.4. Sum and product 28

2.5. Homomorphism modules 30

2.6. Tensor product modules 33

2.7. Change **of** rings 36

3. Exact sequences **of** modules 39

3.1. Exact sequences 39

3.2. The snake lemma 43

3.3. Exactness **of** Hom 48

3.4. Exactness **of** Tensor 49

3.5. Projective modules 50

3.6. Injective modules 52

3.7. Flat modules 54

4. Fraction constructions 57

4.1. Rings **of** fractions 57

4.2. Modules **of** fractions 58

4.3. Exactness **of** fractions 60

4.4. Tensor modules **of** fractions 62

4.5. Homomorphism modules **of** fractions 63

4.6. The polynomial ring is factorial 64

5. Localization 65

5.1. Prime ideals 65

5.2. Localization **of** rings 67

5

6 CONTENTS

5.3. Localization **of** modules 68

5.4. Exactness and localization 70

5.5. Flat ring homomorphisms 71

6. Finite modules 73

6.1. Finite Modules 73

6.2. Free Modules 75

6.3. Cayley-Hamilton’s theorem 77

6.4. Nakayama’s Lemma 78

6.5. Finite Presented Modules 80

6.6. Finite ring homomorphisms 83

7. Modules **of** finite length 85

7.1. Simple Modules 85

7.2. The Length 85

7.3. Artinian Rings 87

7.4. Localization 90

7.5. Local artinian ring 91

8. Noetherian modules 93

8.1. Modules and submodules 93

8.2. Noetherian rings 94

8.3. Finite type rings 95

8.4. Power series rings 97

8.5. Fractions and localization 98

8.6. Prime filtrations **of** modules 98

9. Primary decomposition 101

9.1. Support **of** modules 101

9.2. Ass **of** modules 103

9.3. Primary modules 106

9.4. Decomposition **of** modules 106

9.5. Decomposition **of** ideals 108

10. Dedekind rings 111

10.1. Principal ideal domains 111

10.2. Discrete valuation rings 112

10.3. Dedekind domains 113

Bibliography 117

Index 119

Prerequisites

The basic notions from **algebra**, such as groups, rings, fields and their homomorphisms

together with some linear **algebra**, bilinear forms, matrices and determinants.

Linear **algebra**: Fraleigh & Beauregard, Linear **algebra**, New York 1995.

Algebra: Niels Lauritzen, Concrete abstract **algebra**, Cambridge 2003.

Also recommended: Jens Carsten Jantzen, Algebra 2, Aarhus 2004.

The propositions are stated complete and precise, while the pro**of**s are quite short.

No specific references to the literature are given. But lacking details may all be

found at appropriate places in the books listed in the bibliography.

Nielsen, University **of** Aarhus, Winter 2004

7

1

A dictionary on rings and ideals

1.1. Rings

1.1.1. Definition. An abelian group is a set A with an addition A×A → A, (a, b) ↦→

a + b and a zero 0 ∈ A satisfying

(1) associative: (a + b) + c = a + (b + c)

(2) zero: a + 0 = a = 0 + a

(3) negative: a + (−a) = 0

(4) commutative: a + b = b + a

for all a, b, c ∈ A. A subset B ⊂ A is a subgroup if 0 ∈ B and a − b ∈ B for

all a, b ∈ B. The factor group A/B is the abelian group whose elements are the

cosets a + B = {a + b|b ∈ B} with addition (a + B) + (b + B) = (a + b) + B. A

homomorphism **of** groups φ : A → C respects addition φ(a + b) = φ(a) + φ(b).

The projection π : A → A/B, a ↦→ a + B is a homomorphism. If φ(b) = 0 for all

b ∈ B, then there is a unique homomorphism φ ′ : A/B → C such that φ = φ ′ ◦ π.

1.1.2. Definition. A ring is an abelian group R, addition (a, b) ↦→ a + b and zero

0, together with a multiplication R × R → R, (a, b) ↦→ ab and an identity 1 ∈ R

satisfying

(1) associative: (ab)c = a(bc)

(2) distributive: a(b + c) = ab + ac, (a + b)c = ac + bc

(3) identity : 1a = a = a1

(4) commutative : ab = ba

for all a, b, c ∈ R. If (4) is not satisfied then R is a noncommutative ring. A

subring R ′ ⊂ R is an additive subgroup such that 1 ∈ R ′ and ab ∈ R ′ for all

a, b ∈ R ′ . The inclusion R ′ ⊂ R is a ring extension. A homomorphism **of** rings

φ : R → S is an additive group homomorphism respecting multiplication and

identity

φ(a + b) = φ(a) + φ(b), φ(ab) = φ(a)φ(b), φ(1) = 1

An isomorphism is a homomorphism φ : R → S having an inverse map φ −1 :

S → R which is also a homomorphism. The identity isomorphism is denoted

1R : R → R.

1.1.3. Remark. (1) A bijective ring homomorphism is an isomorphism.

(2) Recall the usual formulas: a + (−b) = a − b, 0a = 0, (−1)a = −a.

(3) The identity 1 is unique.

(4) A ring R is nonzero if and only if the elements 0 = 1.

(5) If φ : R → S is a ring homomorphism, then φ(0) = 0 and φ(R) is a subring

**of** S.

(6) The unique additive group homomorphism Z → R, 1 ↦→ 1 is a ring homomorphism.

9

10 1. A DICTIONARY ON RINGS AND IDEALS

1.1.4. Proposition. Let R1, R2 be rings. The product ring is the product **of** additive

groups R1×R2, ((a1, a2), (b1, b2)) ↦→ (a1+b1, a2+b2), with coordinate multiplication

((a1, a2), (b1, b2)) ↦→ (a1b1, a2b2). The element (1, 1) is the identity. The

projections R1 × R2 → R1, (a1, a2) ↦→ a1 and R1 × R2 → R2, (a1, a2) ↦→ a2 are

ring homomorphisms.

1.1.5. Lemma. In a ring R the binomial formula is true

(a + b) n n

n

= a

k

n−k b k

a, b ∈ R and n a positive integer.

k=0

Pro**of**. The multiplication is commutative, so the usual pro**of** for numbers works.

Use the binomial identity

n

+

k − 1

together with induction on n.

n

=

k

n + 1

1.1.6. Definition. a ∈ R is a nonzero divisor if ab = 0 for all b = 0 otherwise a

zero divisor. a is a unit if there is a b such that ab = 1.

1.1.7. Remark. (1) A unit is a nonzero divisor.

(2) If ab = 1 then b is uniquely determined by a and denoted b = a −1 .

1.1.8. Definition. A nonzero ring R is a domain if every nonzero element is a

nonzero divisor and a field if every nonzero element is a unit. Clearly a field is a

domain.

1.1.9. Example. The integers Z is a domain. The units in Z are {±1}. The rational

numbers Q, the real numbers R and the complex numbers C are fields. The natural

numbers N is not a ring.

1.1.10. Example. The set **of** n × n-matrices with entries from a commutative ring

is an important normally noncommutative ring.

1.1.11. Exercise. (1) Show that the product **of** two domains is never a domain.

(2) Let R be a ring. Show that the set **of** matrices

a

U2 =

0

a,

b

b ∈ R

a

with matrix addition and matrix multiplication is a ring.

(3) Show that the set **of** matrices with real number entries

a,

a −b

b ∈ R

b a

with matrix addition and multiplication is a field isomorphic to C.

(4) Show that the composition **of** two ring homomorphisms is again a ring homomorphism.

(5) Show the claim 1.1.3 that a bijective ring homomorphism is a ring isomorphism.

(6) Let φ : 0 → R be a ring homomorphism from the zero ring. Show that R is itself the

zero ring.

k

1.2. IDEALS 11

1.2. Ideals

1.2.1. Definition. Let R be a ring. An ideal I is an additive subgroup **of** R such

that ab ∈ I for all a ∈ R, b ∈ I. A proper ideal is an ideal I = R.

1.2.2.

Lemma. Let {Iα} be a family **of** ideals in R. Then the additive subgroups

α Iα and

α Iα are ideals.

Pro**of**. The claim for the intersection is clear. Use the formulas bα + cα =

(bα + cα) and a bα = abα to conclude the claim for the sum.

1.2.3. Definition. The intersection, 1.2.2, **of** all ideals containing a subset B ⊂ R

is the ideal generated by B and denoted (B) = RB = BR. It is the smallest ideal

containing B. A principal ideal (b) = Rb is an ideal generated by one element. A

finite ideal (b1, . . . , bn) is an ideal generated by finitely many elements. The zero

ideal is (0) = {0}. The ring itself is a principal ideal, (1) = R. The ideal product

**of** two ideals I, J is denoted IJ and is the ideal generated by all ab, a ∈ I, b ∈ J.

This generalizes to the product **of** finitely many ideals. The power **of** an ideal is

denoted I n . The colon ideal I : J is the ideal **of** elements a ∈ R such that aJ ⊂ I.

1.2.4. Example. (1) Every ideal in Z is principal.

(2) In a field (0), (1) are the only ideals.

(3) A subring is normally not an ideal.

(4) Let K be a field. In K × K there are four ideals (0), (1), ((1, 0)), ((0, 1)).

1.2.5. Proposition. Let R be a ring and B a subset, then RB =

b∈B Rb

A principal ideal is

A finite ideal is

RB = {a1b1 + · · · + anbn|ai ∈ R, bi ∈ B}

Rb = {ab|a ∈ R}

(b1, . . . , bn) = Rb1 + · · · + Rbn

Pro**of**. The righthand side is contained in the ideal RB. Moreover the righthand

side is an ideal containing B, so equality.

1.2.6. Definition. Let φ : R → S be a ring homomorphism. For an ideal J ⊂ S

the contracted ideal is φ −1 (J) ⊂ R and denoted J ∩ R. The kernel is the ideal

Ker φ = φ −1 (0). For an ideal I ⊂ R the extended ideal is the ideal φ(I)S ⊂ S

and denoted IS. Note that (J ∩ R)S ⊂ J and I ⊂ (IS) ∩ R.

1.2.7. Lemma. Let I ⊂ R be an ideal and let R/I be the additive factor group.

The multiplication

R/I × R/I → R/I, (a + I, b + I) ↦→ ab + I

is well defined. Together with the addition the conditions **of** 1.1.2 are satisfied.

Pro**of**. If a + I = a ′ + I and b + I = b ′ + I then a − a ′ , b − b ′ ∈ I and so

ab − a ′ b ′ = a(b − b ′ ) + b ′ (a − a ′ ) ∈ I. Therefore ab + I = a ′ b ′ + I and the

multiplication is well defined. Clearly 1.1.2 are satisfied.

1.2.8. Definition. Let R be a ring and I an ideal, then the factor ring is the additive

factor group R/I with addition (a + I, b + I) ↦→ (a + b) + I and multiplication,

1.2.7, (a + I, b + I) ↦→ ab + I. The projection π : R → R/I, a ↦→ a + I is a ring

homomorphism.

12 1. A DICTIONARY ON RINGS AND IDEALS

1.2.9. Proposition. Let φ : R → S be a ring homomorphism.

(1) Let I ⊂ Ker φ be an ideal. Then there is a unique ring homomorphism

φ ′ : R/I → S such that φ = φ ′ ◦ π.

R

π

R/I

φ

(2) The homomorphism φ ′ : R/ Ker φ → S is a ring isomorphism onto the

subring φ(R) **of** S.

R

π

R/ Ker φ

φ

φ ′

φ ′

S

φ(R)

(3) For any ideal J ⊂ S, I = φ −1 (J) ⊂ R is an ideal and the map φ ′ : R/I →

S/J is an injective ring homomorphism.

Pro**of**. The statements are clear for the addition and the factor map φ ′ (a + I) =

φ(a) is clearly a ring homomorphism.

1.2.10. Corollary. Let π : R → R/I be the projection. The map I ′ ↦→ J =

π −1 (I ′ ) gives a bijective correspondence between ideals I ′ in R/I and ideals J in

R containing I. Also I ′ = π(J) = J/I. This correspondence preserves inclusions,

sums and intersections **of** ideals.

1.2.11. Corollary. Let I ⊂ J ⊂ R be ideals. Then there is a canonical isomorphism

R/J → (R/I)/(J/I)

Pro**of**. The kernel **of** the surjective east-south composite

R

R/J

R/I

(R/I)/(J/I)

is J. By 1.2.9 the horizontal lower factor map gives the isomorphism.

1.2.12. Example. For any integer n the ideals in the factor ring Z/(n) correspond

to ideals (m) ⊂ Z where m divides n.

1.2.13. Definition. Let R be a ring. The additive kernel **of** the unique ring homomorphism

Z → R is a principal ideal generated by a natural number char(R), the

characteristic **of** R. Z/(char(R)) is isomorphic to the smallest subring **of** R.

1.2.14. Proposition. If the characteristic char(R) = p is a prime number, then

the Frobenius homomorphism R → R, a ↦→ a p is a ring homomorphism.

Pro**of**. By the binomial formula 1.1.4

(a + b) p p

p

=

k

k=0

a p−k b k = a p + b p

1.3. PRIME IDEALS 13

since a prime number p divides p

k , 0 < k < p. Clearly (ab) p = apbp .

1.2.15. Exercise. (1) Let I, J be ideals in R. Show that the ideal product

IJ = {a1b1 + · · · + anbn|ai ∈ I, bi ∈ J}

(2) Let I ⊂ R be an ideal. Show that I = I : R.

(3) Show that a ∈ R is a unit if and only if (a) = R.

(4) Show that a ring is a field if and only if (0) = (1) are the only two ideals.

(5) Show that a nonzero ring K is a field if and only if any nonzero ring homomorphism

φ : K → R is injective.

(6) Let m, n be natural numbers. Determine the ideals in Z

(m, n), (m) + (n), (m) ∩ (n), (m)(n)

as principal ideals.

(7) Show that a additive cyclic group has a unique ring structure.

(8) Let p be a prime number. What is the Frobenius homomorphism on the ring Z/(p)?

1.3. Prime ideals

1.3.1. Definition. Let R be a ring and P = R a proper ideal.

(1) P is a prime ideal if for any product ab ∈ P either a ∈ P or b ∈ P . This

amounts to: if a, b /∈ P then ab /∈ P .

(2) P is a maximal ideal if no proper ideal = P contains P .

1.3.2. Proposition. Let P be a prime ideal and I1, . . . In ideals such that I1 . . . In ⊂

P , then some Ik ⊂ P .

Pro**of**. If there exist ak ∈ Ik\P for all k, then since P is prime a1 . . . an ∈

I1 . . . In\P contradicting the inclusion I1 . . . In ⊂ P .

1.3.3. Proposition. Let R be a ring and P an ideal.

(1) P is a prime ideal if and only if R/P is a domain.

(2) P is a maximal ideal if and only if R/P is a field.

Pro**of**. Remark P = R ⇔ R/P = 0. (1) Assume a + P, b + P are nonzero in

R/P . Then a, b /∈ P . So if P is prime then by 1.3.1 ab /∈ P and ab + P is nonzero

in R/P . It follows that R/P is a domain. The converse is similar.

(2) Assume R/P is a field and a /∈ P . Then a + P is a unit in R/P and there is

b such that ab − 1 ∈ P . It follows that the ideal (a) + P = R and therefore P is

maximal. The converse is similar.

1.3.4. Corollary. (1) A maximal ideal is a prime ideal.

(2) A ring is an domain if and only if the zero ideal is a prime ideal.

(3) A ring a field if and only if the zero ideal is a maximal ideal.

1.3.5. Corollary. (1) If φ : R → S is a ring homomorphism and Q ⊂ S a prime

ideal then φ −1 (Q) is a prime ideal **of** R.

(2) Let I ⊂ R be an ideal. An ideal I ⊂ P is a prime ideal in R if and only if

P/I is a prime ideal in R/I.

(3) Let I ⊂ R be an ideal. An ideal I ⊂ P is a maximal ideal in R if and only if

P/I is a maximal ideal in R/I.

Pro**of**. (1) By 1.2.9 R/φ −1 (Q) is a subring **of** the domain S/Q. (2) (3) By 1.2.11

R/P (R/I)/(P/I).

14 1. A DICTIONARY ON RINGS AND IDEALS

1.3.6. Example. An ideal in Z is a prime ideal if it is generated by 0 or a prime

number. Any nonzero prime ideal is a maximal ideal.

1.3.7. Definition. For an ideal I in a ring R the radical is

√ I = {a ∈ R|a n ∈ I for some n}

a is nilpotent is a n = 0 for some positive integer n. A ring is reduced if the

nilradical √ 0 = 0, that is if 0 is the only nilpotent element.

1.3.8. Proposition. (1) The radical **of** an ideal is an ideal.

(2) The nilradical is contained in any prime ideal.

(3) A domain is reduced.

Pro**of**. (1) If am , bn ∈ I then by the binomial formula

(a + b) m+n m+n

m + n

=

a

k

m+n−k b k ∈ I

k=0

and the radical is an ideal. (2) (3) Clearly a nilpotent element is contained in any

prime ideal.

1.3.9. Exercise. (1) Show that the characteristic **of** a domain is either 0 or a prime

number.

(2) Let m, n be a natural numbers. Show that n + (m) ∈ Z/(m) is a unit if and only if

m, n are relative prime.

(3) Let m be a natural number. Show that Z/(m) is reduced if m is square free.

(4) Show that a product **of** reduced rings is reduced.

(5) Let a be nilpotent. Show that 1 − a is a unit.

(6) Let I, J be ideals. Show that √ IJ = √ I ∩ J = √ I ∩ √ J.

(7) Assume an ideal I is contained in a prime ideal P . Show that √ I ⊂ P .

1.4. Chinese remainders

1.4.1. Definition. Ideals I, J ⊂ R are comaximal ideals if I + J = R.

1.4.2. Proposition (Chinese remainder theorem). Let I1, . . . , Ik be pairwise comaximal

ideals in a ring R. Then

(1) For a1, . . . , ak ∈ R there is a a ∈ R, such that a−am ∈ Im for m = 1, . . . , k

(2)

(3) The product **of** projections

is an isomorphism.

Pro**of**. (1) For each m

I1 · · · Ik = I1 ∩ · · · ∩ Ik

R/I1 · · · Ik → R/I1 × · · · × R/Ik

R =

(Im + In) = Im +

n=m

n=m

So choose um ∈ Im and vm ∈

n=m In with um + vm = 1. Put a = a1v1 + · · · +

akvk. Then a − am = · · · + amum + · · · ∈ Im. (2) For a in the intersection assume

by induction that a ∈ I2 · · · Ik. From the pro**of** **of** (1) a = u1a + av1 ∈ I1 · · · Ik.

(3) Subjectivity follows from (1). The kernel is the intersection which by (2) is the

product. 1.2.9 gives the isomorphism.

In

1.5. UNIQUE FACTORIZATION 15

1.4.3. Corollary. Let P1, . . . , Pk be pairwise different maximal ideals in a ring R.

Then

and

P n1

1 · · · P nk n1

nk

k = P1 ∩ · · · ∩ Pk R/P n1

1 · · · P nk

n1

nk

k → R/P1 × · · · × R/Pk is an isomorphism.

1.4.4. Definition. An element e in a ring R is idempotent if e = e 2 . A nontrivial

idempotent is an idempotent e = 0, 1.

1.4.5. Proposition. A ring R is a product **of** two nonzero rings if and only if it

contains a nontrivial idempotent e.

Pro**of**. Use that the ideals Re and R(1 − e) are proper and comaximal.

1.4.6. Exercise. (1) Show that for a prime number p the rings Z/(p2 ) and Z/(p) ×

Z/(p) are not isomorphic.

(2) Let n = p n1

1 be a factorization into different primes. Show that

. . . pnk

k

Z/(p n1

1

· · · pnk

k

) → Z/(pn1 1 ) × · · · × Z/(pnk k )

is an isomorphism.

(3) Let elements e1 + e2 = 1 with e1e2 = 0 be given in a ring R. Show that R

R/(e1) × R/(e2).

(4) Let I, J be ideals such that √ I, √ J are comaximal. Show that I, J are comaximal.

1.5. Unique factorization

1.5.1. Lemma. Let R be a domain and (a) = (b) principal ideals. Then there is a

unit u ∈ R such that b = ua.

Pro**of**. b = ua and a = vb giving b = uvb. If b = 0 then uv = 1 showing that u is

a unit.

1.5.2. Definition. Let R be a domain and P the set **of** principal ideals different

from (0) and R. By 1.5.1 multiplication **of** generators gives a well defined multiplication

**of** principal ideals on P. An element in P maximal for inclusion is an

irreducible principal ideal. A generator **of** an irreducible element is an irreducible

element in R.

1.5.3. Definition. A domain R is a unique factorization domain if

(1) Every irreducible element in P is a prime ideal.

(2) Every element in P is a product **of** irreducible elements.

1.5.4. Proposition. In a unique factorization domain the factorization **of** elements

in P into irreducibles is unique up to order.

Pro**of**. Proceed by induction on the shortest factorization **of** an element in P. Let

(a1) . . . (am) = (b1) . . . (bn) be factorizations into irreducibles. (a1) is a prime

ideal, so by 1.3.2 and reordering (b1) = (a1). By cancellation m − 1 = n − 1 and

(ai) = (bi) after a reordering.

1.5.5. Definition. A domain R is a principal ideal domain if every ideal is principal.

1.5.6. Proposition. A principal ideal domain R is a unique factorization domain.

16 1. A DICTIONARY ON RINGS AND IDEALS

Pro**of**. Let (a) be irreducible and x, y /∈ (a). Then (a, x), (a, y) are principal ideals

properly containing (a) giving (a, x) = (a, y) = R. Let ba + cx = da + ey = 1

and look at (ba+cx)(da+ey) = 1 to see that xy /∈ (a). It follows that (a) is prime,

part (1) **of** 1.5.3. If (b) is not irreducible, then (b) = (a1)(b1) for some irreducible

(a1) and (b) ⊂ (b1). Continue this process to get (b) = (a1) . . . (an)(bn) for some

irreducibles (a1), . . . (an). The chain **of** ideals (b) ⊂ (b1) ⊂ · · · ⊂ (bn) has a

union

n (bn) which is a principal ideal. The generator **of** the union must be in

some (bn). Therefore (bn) = (bn+1) giving that (bn) is irreducible. This gives a

factorization required in 1.5.3 part (2).

1.5.7. Example. The integers Z is a principal ideal domain and therefore a unique

factorization domain.

1.5.8. Definition. The supremum **of** a set **of** elements in P is the greatest common

divisor and an infimum is the least common multiple.

1.5.9. Corollary. In a unique factorization domain the greatest common divisor

and the least common multiple **of** a finite set **of** elements exist.

If (a) = (p m1

1 ) . . . (pmk k ) and (b) = (pn1 1 ) . . . (pnk k ) with mi, ni ≥ 0 then greatest

common divisor is (p min(m1,n1)

1 ) . . . (p min(mk,nk)

k ) and least common multiple is

).

(p max(m1,n1)

1

) . . . (p max(mk,nk)

k

1.5.10. Exercise. (1) Show that an irreducible element in a principal ideal domain generates

a maximal ideal.

(2) Show that there are infinitely many prime numbers.

(3) Let Z[ √ −1] be the smallest subring **of** C containing √ −1. Show that Z[ √ −1] is a

principal ideal domain.

(4) Let Z[ √ −5] be the smallest subring **of** C containing √ −5. Show that Z[ √ −5] is not

a unique factorization domain.

1.6. Polynomials

1.6.1. Definition. Let R be a ring. The polynomial ring R[X] is the additive

group given by the direct sum

n RXn , n = 0, 1, 2, . . . consisting **of** all finite

sums f = a0 + a1X + . . . amX m , a polynomial with an ∈ R being the n ′ th

coefficient. Multiplication is given by XiX j = Xi+j extended by linearity. If

g = b0 + b1X + . . . bnX n is an other polynomial, then

f + g = (a0 + b0) + (a1 + b1)X + · · · + (ak + bk)X k + . . .

fg = a0b0 + (a0b1 + a1b0)X + · · · + (a0bk + a1bk−1 + · · · + akb0)X k + . . .

A monomial is polynomial **of** form aX n . The construction may be repeated to

give the polynomial ring in n-variables R[X1, . . . , Xn] or even in infinitely many

variables.

1.6.2. Definition. The degree, deg(f), **of** a polynomial 0 = f ∈ R[X] is the

index **of** the highest nonzero coefficient, the leading coefficient. A polynomial

with leading coefficient the identity is a monic polynomial.

1.6.3. Remark. (1) R is identified with the subring **of** constants in the polynomial

ring R[X1, . . . , Xn].

(2) The nonzero constants are the polynomials **of** degree 0.

(3) The constant polynomial 1 is the unique monic polynomial **of** degree 0 and

the identity in the polynomial ring.

1.6. POLYNOMIALS 17

1.6.4. Proposition. Let 0 = f, g ∈ R[X].

(1) If fg = 0 then deg(fg) ≤ deg(f) + deg(g).

(2) If the leading coefficient **of** f or g is a nonzero divisor in R, then fg = 0 and

deg(fg) = deg(f) + deg(g)

Pro**of**. (1) This is clear. (2)Clearly the leading coefficient **of** the product is the

product **of** the leading coefficients.

1.6.5. Corollary. Let R be a domain.

(1) The polynomial ring R[X] is a domain.

(2) The units in R[X] are the constants, which are units in R.

1.6.6. Proposition. Let R be a domain, 0 = f, d ∈ R[X] polynomials with d

monic. Then there are a unique q, r ∈ R[X] such that

f = qd + r, r = 0 or deg(r) < deg(d)

Pro**of**. Induction on deg(f). If deg(f) < deg(d) then q = 0, r = f. Otherwise

if a is the leading coefficient **of** f, then f − adX deg(f)−deg(d) has degree less than

deg(f). By induction f − adX deg(f)−deg(d) = qd + r giving the claim.

1.6.7. Proposition. Let φ : R → S be a ring homomorphism. For any element

b ∈ S there is a unique ring homomorphism R[X] → S extending φ and mapping

X ↦→ b.

Pro**of**.

a0 + a1X + . . . amX m ↦→ φ(a0) + φ(a1)b + . . . φ(am)b m

is clearly the one and only choice.

1.6.8. Definition. The homomorphism in 1.6.7 is the evaluation map at b in S.

The image **of** a polynomial f ∈ R[X] is denoted f(b) ∈ S.

1.6.9. Proposition. Let I ⊂ R be an ideal. Then there is a canonical isomorphism.

(R/I)[X] R[X]/IR[X]

Pro**of**. There is an obvious pair **of** inverse homomorphisms constructed by 1.2.9

and 1.6.7.

1.6.10. Corollary. If P ⊂ R is a prime ideal, then P R[X] ⊂ R[X] is a prime

ideal.

1.6.11. Definition. Let φ : R → S be a ring homomorphism and B ⊂ S a subset.

The ring generated over R by B is

R[B] = φ(R)[B] ⊂ S

the smallest subring **of** S containing φ(R) ∪ B. If there is a finite subset B such

that R[B] = S then S is a finite type ring or a finitely generated ring over R.

1.6.12. Corollary. Let φ : R → S be a ring homomorphism.

(1) If bα ∈ S and Xα is a family **of** variables, then there is a surjective ring

homomorphism

R[Xα] → R[bα], Xα ↦→ bα

making R[bα] a factor ring **of** the polynomial ring R[Xα].

18 1. A DICTIONARY ON RINGS AND IDEALS

(2) If S is a finite type ring over R, then S is a factor ring **of** a polynomial ring

in finitely many variables over R.

1.6.13. Exercise. (1) Let K be a field. Show that there are infinitely many prime ideals

in K[X].

(2) What are the units in the ring Z[X]/(1 − 2X)?

(3) Determine the prime ideals in Q[X]/(X − X 2 ).

(4) Show that the ring Z[X] is not a principal ideal domain.

(5) Show that the ring Q[X, Y ] is not a principal ideal domain.

1.7. Roots

1.7.1. Definition. Let φ : R → S be a ring homomorphism and f ∈ R[X] a

polynomial. An element b ∈ S is a root **of** f (in S) if the evaluation f(b) = 0.

1.7.2. Proposition. Let R be a domain. An element a ∈ R is a root **of** the polynomial

f ∈ R[X] if and only if there is a q ∈ R[X] such that

f = q(X − a)

Pro**of**. By 1.6.6 f = q(X −a)+r. It follows that a is a root if and only if r = 0.

1.7.3. Corollary. Let R is a domain. There are at most deg(f) roots in a nonzero

polynomial f ∈ R[X].

1.7.4. Definition. The multiplicity **of** a root a **of** a nonzero polynomial f ∈ R[X]

is highest m such that

f = q(X − a) m

A root **of** multiplicity m = 1 is a simple root.

1.7.5. Corollary. Let R is a domain. If m1, . . . , mk are the multiplicities **of** the

roots **of** a nonzero polynomial f ∈ R[X], then m1 + · · · + mk ≤ deg(f).

1.7.6. Definition. The derivative **of** a polynomial f = anX n ∈ R[X] is

1.7.7. Lemma. The derivative satisfies

(1) (f + g) ′ = f ′ + g ′ .

(2) (fg) ′ = f ′ g + fg ′

(3) If f is constant, then f ′ = 0.

f ′ = nanX n−1

1.7.8. Proposition. Let R is a domain. An element a ∈ R is a root **of** multiplicity

m > 1 **of** a nonzero f ∈ R[X] if and only if a is a root **of** f and f ′ .

Pro**of**. By 1.6.6 f = q(X − a) 2 + cX + d and by 1.7.7 f ′ = q ′ (X − a) 2 + 2q(X −

a) + c. I follows that a is a root **of** multiplicity m > 1 if and only if c = d = 0.

1.7.9. Exercise. (1) Let a1, . . . ak be roots with multiplicities m1, . . . , mk in a polynomial

f. Show that m1 + · · · + mk ≤ deg(f).

(2) Let K be a field and let a1, . . . , an ∈ K. Show that the ideal (X1 −a1, . . . , Xn −an)

is maximal in K[X1, . . . , Xn].

(3) Let the characteristic char(R) = n > 0. What is (X n ) ′ in R[X].

1.8. FIELDS 19

1.8. Fields

1.8.1. Definition. Let p ∈ Z be a prime number. The factor ring Fp = Z/(p) is a

field with p elements. Together with Q they constitute the prime fields.

1.8.2. Proposition. Let K be a field then the polynomial ring K[X] is a principal

ideal domain.

Pro**of**. Let d = 0 be a polynomial **of** lowest degree in an ideal I. Given f ∈ I then

by 1.6.6 f = qd + r with r = f − qd ∈ I. By degree considerations r = 0 and

I = (d).

1.8.3. Corollary. Let K be a field then the polynomial ring K[X] is a unique

factorization domain.

Pro**of**. Follows from 1.5.6.

1.8.4. Definition. A subfield is a subring, which is a field. A field extension is the

inclusion **of** a subfield K ⊂ L in a field. A finite field extension K ⊂ L is an

extension, where L is finite dimensional as a vector space over K.

1.8.5. Example. (1) Let K be a field and f an irreducible polynomial in K[X].

Then K ⊂ K[X]/(f) is a finite field extension.

(2) Let K ⊂ R ⊂ L be a subring in a finite field extension. Then R is a field.

Namely multiplication on R with a nonzero element **of** R is a K-linear map

on the finite dimensional K-vector space R and therefore an isomorphism.

1.8.6. Proposition. (1) Let K be a field and f a polynomial in K[X]. Then there

is a finite field extension K ⊂ L such that f factors in linear factors in L[X].

(2) If K ⊂ L1 and K ⊂ L2 are finite field extensions then there is a finite field

extension K ⊂ L such that L1 ∪ L2 ⊂ L.

Pro**of**. (1) Assume f irreducible. In L = K[X]/(f) the class X + (f) is a root **of**

f. In general proceed adjoining roots **of** irreducible factors **of** f. (2) An element x

in a finite field extension K ⊂ K ′ is the root **of** the irreducible monic polynomial

f generating the kernel **of** the evaluation homomorphism K[X] → K ′ , X ↦→ x.

Now proceed by (1) adjoining elements in L2 to L1.

1.8.7. Proposition. Let p ∈ Z be a prime number. For any power q = p n there is

a field Fq with q elements, unique up to isomorphism.

Pro**of**. Let Fp ⊂ K be a field extension, where X q − X factors into linear factors,

1.8.6 (1). The subset **of** roots is the set **of** elements fixed under n-times the

Frobenius and therefore a subring being a subfield by 1.8.5 (2). The derivative

(X q − X) ′ = −1 so by 1.7.8 there are q elements in this subfield. Uniqueness

follows from 1.8.6 (2).

1.8.8. Exercise. (1) Show that the ring R[X]/(X 2 + 1) is isomorphic to the field **of**

complex numbers.

(2) Show that the ring F2[X]/(X 2 + X + 1) is a field with 4 elements.

(3) Show that the ring F2[X]/(X 3 + X + 1) is a field with 8 elements.

(4) Let K ⊂ L be a field extension **of** fields **of** characteristic 0 and let a ∈ L be a root **of**

an irreducible polynomial f ∈ K[X]. Show that a is a simple root.

(5) Let p be a prime number. Show that Fp is the only ring with p elements.

(6) Let p be a prime number. Show that a ring with p 2 elements is isomorphic to one **of**

four non isomorphic Z/(p 2 ), Fp × Fp, Fp[X]/(X 2 ), F p 2.

20 1. A DICTIONARY ON RINGS AND IDEALS

1.9. Power series

1.9.1. Definition. Let R be a ring. The power series ring R[[X]] is the additive

group

n RXn , n = 0, 1, 2, . . . **of** all power series anX n with n ′ th coefficient

an ∈ R. Multiplication is given by X i X j = X i+j extended by linearity. For

another power series bnX n the sum and product are

anX n + bnX n = (an + bn)X n

anX n · bnX n = (

k

an−kbk)X n

The construction may be repeated to give the power series ring in n-variables

R[[X1, . . . , Xn]] or even in infinitely many variables.

1.9.2. Remark. The polynomial ring is identified as a subring R[X] ⊂ R[[X]] **of**

power series with only finitely many nonzero terms.

1.9.3. Definition. The order, o(f), **of** a power series 0 = f ∈ R[[X]] is the index

**of** the least nonzero coefficient.

1.9.4. Proposition. If R is a domain, then R[[X]] is a domain and for 0 = f, g ∈

R[X]

o(fg) = o(f) + o(g)

Pro**of**. Clearly the lowest nonzero coefficient in the product is the product **of** the

two lowest nonzero coefficients.

1.9.5. Proposition. A power series f = anX n is a unit if and only if a0 is a

unit.

Pro**of**. It suffices to look at a power series f = 1 − gX. Then the power series

1/f = 1 + gX + g 2 X 2 + · · · + g n X n + . . . is well defined and f · 1/f = 1.

1.9.6. Proposition. If K is a field, then K[[X]] is a principal ideal domain. and

(X) is the only nonzero prime ideal.

Pro**of**. If the lowest order **of** an element in an ideal I is n. Then clearly I =

(X n ).

1.9.7. Corollary. If K is a field, then K[[X]] is a unique factorization domain.

1.9.8. Proposition. Let I ⊂ R be an ideal. Then there is a canonical surjective

homomorphism

R[[X]]/IR[[X]] → R/I[[X]]

1.9.9. Corollary. If Q ⊂ R[[X]] is a maximal ideal, then P = Q ∩ R ⊂ R is a

maximal ideal and Q = (P, X).

Pro**of**. X ∈ Q so R/Q ∩ R R[[X]]/Q.

1.9.10. Exercise. (1) Show that the ring Z[[X]] is not a principal ideal domain.

(2) Show that the ring Q[[X, Y ]] is not a principal ideal domain.

(3) Let K be a field. Show that (X1, . . . , Xn) is the unique maximal ideal in the power

series ring K[[X1, . . . , Xn]].

(4) Let a ∈ R be nilpotent. Show that the ring R[[X]]/(X − a) is isomorphic to R.

(5) What is R[[X]]/(X − a) if a ∈ R is a unit?

(6) Let I ⊂ R be an ideal. Show that IR[[X]] ⊂ R[[X]] is not a maximal ideal.

2

Modules

2.1. Modules and homomorphisms

2.1.1. Definition. Let R be a ring. A module (R-module) is an abelian group

M, addition (x, y) ↦→ x + y and zero 0, together with a scalar multiplication

R × M → M, (a, x) ↦→ ax satisfying

(1) associative : (ab)x = a(bx)

(2) bilinear : a(x + y) = ax + ay, (a + b)x = ax + bx

(3) identity: 1x = x

for all a, b ∈ R, x, y ∈ M. A submodule M ′ ⊂ M is an additive subgroup such

that ax ∈ M ′ for all a ∈ R, x ∈ M ′ . A homomorphism is an additive group

homomorphism f : M → N respecting scalar multiplication

f(x + y) = f(x) + f(y), f(ax) = af(y)

for all a ∈ R, x, y ∈ M. An isomorphism is a homomorphism f : M → N

having an inverse map f −1 : N → M which is also a homomorphism. The

identity isomorphism is denoted 1M : M → M.

2.1.2. Lemma. Let R be a ring and M a module.

(1) a0 = 0 = 0x

(2) (−1)x = −x

(3) (−a)x = −(ax) = a(−x)

for all a ∈ R, x ∈ M.

Pro**of**. (1) Calculate a0 = a(0 + 0) = a0 + a0 and cancel to get a0 = 0. Similarly

0x = 0. (2) By (1) 0 = 0x = (1 + (−1))x = x + (−1)x, so conclude −x =

(−1)x. (3) Calculate (−a)x = ((−1)a)x = (−1)(ax) = −(ax). Similarly

a(−x) = −(ax).

2.1.3. Lemma. Let R be a ring and f : M → N a homomorphism **of** modules.

(1) f(0) = 0.

(2) f(ax + by) = af(x) + bf(y).

(3) f(−x) = −f(x) for all x ∈ M.

for all a, b ∈ R, x, y ∈ M.

Pro**of**. (1) Calculate f(0) = f(0 + 0) = f(0) + f(0) and conclude f(0) = 0.

(2) Calculate f(ax + by) = f(ax) + f(by) = af(x) + bf(y). (3) By 2.1.2

f(−x) = f((−1)x) = (−1)f(x) = −f(x).

2.1.4. Example. (1) The zero group is the zero module.

(2) Over the zero ring the zero module is the only module.

(3) The zero subgroup **of** a module is the zero submodule.

(4) The ring R is a module under multiplication. An ideal is a submodule.

21

22 2. MODULES

(5) If R is a field, a module is a vector space and a homomorphism is a linear

map.

(6) A module over Z is an abelian group and an additive map **of** abelian groups

is a homomorphism.

2.1.5. Proposition. A bijective homomorphism is an isomorphism.

Pro**of**. Let f : M → N be a bijective homomorphism **of** R-modules and let g :

N → M be the inverse map. For x, y ∈ N write x = f(g(x)), y = f(g(y)) and

get additivity **of** g, g(x + y) = g(f(g(x)) + f(g(y))) = g(f(g(x) + g(y))) =

g(x) + g(y). Similarly for a ∈ R g(ax) = g(af(g(x))) = g(f(ag(x))) = ag(x),

so g respects scalar multiplication and is a homomorphism.

2.1.6. Lemma. Let a ∈ R and M be a module. The map M → M, x ↦→ ax is a

homomorphism.

Pro**of**. Let f(x) = ax and calculate f(x+y) = a(x+y) = ax+ay = f(x)+f(y)

and f(bx) = a(bx) = (ab)x = (ba)x = b(ax) = bf(x) to get that f is a

homomorphism. Remark that the last calculation uses that R is commutative 1.1.2

(4).

2.1.7. Definition. Let a ∈ R and M be a module.

(1) The scalar multiplication with a is the homomorphism, 2.1.6,

aM : M → M, x ↦→ ax

(2) a ∈ R is a nonzero divisor on M if scalar multiplication aM is injective, i.e.

ax = 0 for all 0 = x ∈ M. Otherwise a is a zero divisor.

2.1.8. Remark. The two notions **of** nonzero divisor on R : 1.1.6 as element in the

ring and 2.1.4 as scalar multiplication on the ring coincide.

2.1.9. Example. If R is a field, then scalar multiplication on a vector space is either

zero or an isomorphism.

2.1.10. Lemma. Let φ : R → S be a ring homomorphism and N an S-module.

The map

R × N → N, (a, x) ↦→ φ(a)x

is an R-scalar multiplication on N, 2.1.1.

Pro**of**. Let a, b ∈ R, x, y ∈ N and µ(a, x) = φ(a)x. Calculate µ(a + b, x) =

φ(a + b)x = φ(a)x + φ(b)x = µ(a, x) + µ(b, x), µ(a, x + y) = φ(a)(x + y) =

φ(a)x + φ(a)y = µ(a, x) + µ(a, y), µ(1, x) = φ(1)x = 1x = x and µ(ab, x) =

φ(ab)x = φ(a)φ(b)x = µ(a, µ(bx)) showing the conditions 2.1.1.

2.1.11. Definition. Let φ : R → S be a ring homomorphism. The restriction **of**

scalars **of** an S-module N is the same additive group N viewed as an R-module

through φ. The scalar multiplication is 2.1.10

R × N → N, (a, x) ↦→ ax = φ(a)x

An S-module homomorphism g : N → N ′ is also an R-module homomorphism.

2.1.12. Example. (1) The scalar multiplication with a Restriction **of** scalars for

the unique ring homomorphism Z → R give just the underlying abelian group

**of** a module, 2.1.4 (6).

2.2. SUBMODULES AND FACTOR MODULES 23

(2) Let I ⊂ R be an ideal. Restriction **of** scalars along the projection R → R/I

gives any R/I-module as an R-module.

2.1.13. Proposition. Let R be a ring. There is a dictionary:

(1) To an R[X]-module N associate the pair (N, f) consisting **of** N as R-module

through restriction **of** scalars and f = XN : N → N, f(y) = Xy scalar

multiplication with X as an R-module homomorphism. An R[X]-homomorphism

g : N → N ′ gives an R-homomorphism such that g ◦ f = f ′ ◦ g.

(2) To a pair (N, f) **of** an R-module and a homomorphism f : N → N associate

the R[X]-module with abelian group N and scalar multiplication determined

by Xy = f(y) for y ∈ N. Note

( anX n )y = anf ◦n (y)

An R-homomorphism g : N → N ′ such that g ◦ f = f ′ ◦ g is an R[X]homomorphism.

Pro**of**. The statement is an algorithm to follow.

2.1.14. Proposition. Let R be a ring and M a module. The abelian group R ⊕ M

with multiplication

(a + x)(b + y) = ab + (ay + bx)

is a ring. R is a subring and M is an ideal.

Pro**of**. Simple calculations show that the conditions 1.1.2 are satisfied.

2.1.15. Exercise. (1) Show that a composition **of** homomorphisms is a homomorphism.

(2) Show that composition **of** scalar multiplications with a, b ∈ R on a module M is a

scalar multiplication with the product, aM ◦ bM = (ab)M .

(3) Let φ : R → S be a ring homomorphism. Show that φ is an R-module homomorphism,

when S is viewed as R-module through restriction **of** scalars 2.1.11.

(4) Fill out the dictionary 2.1.13.

2.2. Submodules and factor modules

2.2.1. Lemma. Let R be a ring and M a module. Let Nα be a family **of** submodules.

Then the additive subgroups

α Nα and

α Nα are submodules.

Pro**of**. Use the formulas xα + yα = (xα + yα) and a xα = axα to

conclude that Nα is a submodule. If x, y ∈ Nα for all α, then x + y, ax ∈ Nα

for all α, so Nα is a submodule.

2.2.2. Definition. Let R be a ring and M a module. The intersection **of** all submodules

containing a subset Y ⊂ M is the submodule generated by Y and denoted

RY . This is the smallest submodule, 2.2.1, **of** M containing Y . The module M is

generated by Y if RY = M. Let I be an ideal. The submodule generated by all

products ax, a ∈ I, x ∈ M is denoted IM.

2.2.3. Proposition. Let R be a ring and M a module. If Y ⊂ M, then RY =

y∈Y Ry,

RY = {a1y1 + · · · + anyn|ai ∈ R, yi ∈ Y }

Pro**of**. The righthand side is contained in the submodule RY . Moreover the righthand

side is a submodule containing Y , so equality.

2.2.4. Corollary. Let I ⊂ R be an ideal and M a module.

24 2. MODULES

(1)

IM = {a1y1 + · · · + anyn|ai ∈ I, yi ∈ M}

(2) If I = (a) is principal, then

aM = (a)M = {ay|y ∈ M}

Pro**of**. (1) is clear. (2) By (1) an element in aM is biayi = a biyi = ay for

y = biyi ∈ M as claimed.

2.2.5. Lemma. Let R be a ring, M a module and N ⊂ M a submodule. Let M/N

be the abelian factor group, then the map

R × M/N → M/N, (a, x + N) ↦→ ax + N

is well defined and a scalar multiplication, 2.1.1.

Pro**of**. If x+N = y+N then x−y ∈ N and so a(x−y) = ax−ay ∈ N. Therefore

ax + N = ay + N and the multiplication is well defined. Since representatives

may be chosen such that (x + N) + (y + N) = x + y + N, a(x + N) = ax + N,

the laws for scalar multiplication are satisfied.

2.2.6. Definition. Let R be a ring, M a module and N ⊂ M a submodule, then the

factor module is the additive factor group M/N with, 2.2.5, scalar multiplication

a(x + N) = ax + N. The projection p : M → M/N, x ↦→ x + N is a surjective

homomorphism.

2.2.7. Lemma. Let R be a ring, N ⊂ M a submodule and p : M → M/N

the projection. p is surjective and if Y ⊂ M generates M, then p(Y ) ⊂ M/N

generates the factor module.

Pro**of**. Clearly if RY = M then Rp(Y ) = p(RY ) = M/N.

2.2.8. Example. (1) A submodule **of** R is the same as an ideal.

(2) Both an ideal I ⊂ R and a factor ring R/I are modules.

(3) The module structure on R/I as a factor module and the structure by restriction

**of** scalars through the projection R → R/I are identical.

2.2.9. Proposition. Let R = R1×R2 be the product ring 1.1.4. There is a bijective

(up to natural isomorphism) correspondence.

(1) If M1 is an R1-module and M2 is an R2-module, then M = M1 × M2 is an

R-module with coordinate scalar multiplication. A pair **of** homomorphisms

induce a homomorphism on the product.

(2) If M is an R-module then M1 = (1, 0)M is an R1-module and M2 =

(0, 1)M is an R2-module. A homomorphism induces a pair **of** homomorphisms.

2.2.10. Remark. The correspondence 2.2.9 indicates that the structure **of** modules

and homomorphisms over a product ring is identified with the structure **of** pairs **of**

modules and homomorphisms over each component ring in the product.

2.2.11. Exercise. (1) Give an example **of** two submodules N, L ⊂ M such that the

union N ∪ L is not a submodule.

(2) Let R be a ring and a ∈ R. Show that the R-module R[X]/(X − a) is isomorphic

to R.

(3) Show that the projection p 2.2.6 is a homomorphism.

(4) Fill in the details in the dictionary 2.2.9

2.3. KERNEL AND COKERNEL 25

2.3. Kernel and cokernel

2.3.1. Lemma. Let R be a ring and f : M → N a homomorphism **of** modules.

Given submodules M ′ ⊂ M, N ′ ⊂ N, then f −1 (N ′ ) ⊂ M and f(M ′ ) ⊂ N are

submodules.

Pro**of**. If x, y ∈ f −1 (N ′ ) then f(x + y) = f(x) + f(y) ∈ N ′ and for a ∈ R

f(ax) = af(x) ∈ N ′ so x+y, ax ∈ f −1 (N ′ ) proving f −1 (N ′ ) to be a submodule.

The same equations prove that f(M ′ ) is a submodule.

2.3.2. Definition. Let f : M → N be a homomorphism **of** modules. Then there

are submodules, 2.3.1.

(1) The kernel Ker f = f −1 (0).

(2) The image Im f = f(M).

(3) The cokernel Cok f = N/ Im f.

2.3.3. Proposition. Let f : M → N be a homomorphism **of** modules.

(1) f is injective if and only if Ker f = 0.

(2) f is surjective if and only if Cok f = 0.

(3) f is an isomorphism if and only if Ker f = 0 and Cok f = 0.

Pro**of**. (1) If f is injective and x ∈ Ker f then f(x) = 0 = f(0) so x = 0.

Conversely if Ker f = 0 and f(x) = f(y) then f(x − y) = 0 so x = y. (2) The

factor module N/ Im f = 0 if and only if Im f = N. (3) This follows from (1)

and (2).

2.3.4. Example. Let a ∈ R give scalar multiplication aM : M → M, x ↦→ ax.

(1) Im aM = aM = {ax ∈ M|x ∈ M}.

(2) Ker aM = {x ∈ M|ax = 0}.

(3) Cok aM = M/aM.

2.3.5. Proposition. Let f : M → N be a homomorphism **of** modules.

(1) Let L ⊂ Ker f be a submodule. Then there is a unique homomorphism f ′ :

M/L → N such that f = f ′ ◦ p.

M

p

M/L

f

(2) The homomorphism f ′ : M/ Ker f → N is a module isomorphism onto the

submodule Im f **of** N.

M

p

M/ Ker f

f

f ′

f ′

N

Im f

Pro**of**. (1) If x+L = x ′ +L then x−x ′ ∈ L giving f(x) = f(x ′ ). The factor map

f ′ : M/L → N, x + L ↦→ f(x) is therefore well defined and f = f ′ ◦ p. Since

f, p are homomorphisms and p is surjective it follows that f ′ is a homomorphism.

(2) The kernel **of** f ′ is Ker f/ Ker f = 0 so by 2.3.3 it is an isomorphism.

26 2. MODULES

2.3.6. Corollary. Let p : M → M/N be the projection onto a factor module. The

map L ′ ↦→ L = p −1 (L ′ ) gives a bijective correspondence between submodules

in M/N and submodules in M containing N. Also L ′ = p(L) = L/N. This

correspondence preserves inclusions, additions and intersections **of** submodules.

Pro**of**. If L ′ is a submodule **of** M/N then clearly p(p−1 (L ′ )) = L ′ . If N ⊂ L

is a submodule **of** M then clearly L ⊂ p−1 (p(L)). Moreover if x ∈ p−1 (p(L))

then p(x) = p(y) for some y ∈ L and therefore y − x ∈ N. It follows that

L = p−1 (p(L)) and the correspondence is bijective. Inclusions are easily seen to

be preserved. Also easily p(L1 + L2) = p(L1) + p(L2) and p−1 (L ′ 1 ∩ L′ 2 ) =

p−1 (L ′ 1 ) ∩ p−1 (L ′ 2 ) h**old**. The two resting equalities are consequences **of** this and

bijectivity **of** the correspondence.

2.3.7. Corollary. Let L ⊂ N ⊂ M be submodules. Then there is a canonical

isomorphism

M/N → (M/L)/(N/L)

Pro**of**. The kernel **of** the surjective east-south composite

M

M/N

M/L

(M/L)/(N/L)

is N. By 2.3.5 the horizontal lower factor map gives the isomorphism.

2.3.8. Corollary. Let L, N ⊂ M be submodules. Then there is a canonical isomorphism

N/N ∩ L → N + L/L

given by x + N ∩ L ↦→ x + L.

Pro**of**. The kernel **of** the east-south composite

N

N/N ∩ L

N + L

N + L/L

is N ∩ L. Since x + y + L = x + L for x ∈ N, y ∈ L this composite is also

surjective. By 2.3.5 the horizontal lower factor map gives the isomorphism.

2.3.9. Proposition. Let f : M → N and g : N → L be homomorphisms such that

Im f ⊂ Ker g. Then there is a unique homomorphism g ′ : Cok f → L such that

g = g ′ ◦ p.

Pro**of**. This follows from 2.3.5.

M f

N

g

p

L

g ′

Cok f

2.3.10. Lemma. Let R be a ring and M a module. For x ∈ M the map R →

M, a ↦→ ax is the unique homomorphism such that 1 ↦→ x.

2.3. KERNEL AND COKERNEL 27

Pro**of**. Let f(a) = ax. f(ab) = (ab)x = a(bx) = af(b) shows that f is a

homomorphism. This argument does not use the commutativity 1.1.2 **of** R.

2.3.11. Definition. Denote the homomorphism 2.3.10

The annihilator **of** x is the ideal

1x : R → M, a ↦→ ax

Ann(x) = Ker 1x = {a ∈ R|ax = 0}

For a subset Y ⊂ M the annihilator Ann(Y ) =

y∈Y Ann(y) is the ideal **of**

elements

Ann(Y ) = {a ∈ R|ay = 0, for all y ∈ Y }

2.3.12. Proposition. Let R be a ring and Y a subset **of** a module M.

(1) Ann(Y ) = Ann(RY ).

(2) a ∈ Ann(M) if and only if aM = 0.

(3) If modules M M ′ then Ann(M) = Ann(M ′ ).

(4) The induced homomorphism

is an isomorphism.

1 ′ x : R/ Ann(x) → Rx, a + Ann(x) ↦→ ax

Pro**of**. (1) If a ∈ Ann(Y ) then a biyi = biayi = 0 giving the not so obvious

Ann(Y ) ⊂ Ann(RY ). (2) Clear since aM(x) = ax, 2.1.7. (3) Let f : M → M ′

be an isomorphism and a ∈ R. Then af(x) = f(ax) expresses that f ◦ aM =

aM ′ ◦ f. Since f is bijective, aM = 0 if and only if aM ′ = 0. By (2) Ann(M) =

Ann(M ′ ). (4) This follows from 2.3.5.

2.3.13. Corollary. Let I, J ⊂ R be ideals.

(1) Ann(R/I) = I

(2) If R/I R/J then I = J.

2.3.14. Lemma. Let I ⊂ R be an ideal and M an R-module. If I ⊂ Ann(M)

then M is an R/I-module with the scalar multiplication

That is, M is an R/ Ann(M)-module.

R/I × M → M, (a + I, x) ↦→ ax

2.3.15. Example. Let a ∈ R give scalar multiplication aM : M → M, x ↦→ ax.

(1) a ∈ Ann(Ker aM).

(2) a ∈ Ann(Cok aM).

(3) Ker aM and Cok aM are modules over the factor ring R/(a), 2.3.14.

2.3.16. Definition. Let R be a ring and L, N ⊂ M submodules. The colon ideal

N : L is the ideal **of** elements a ∈ R such that aL ⊂ N.

2.3.17. Proposition. The colon ideal **of** submodules L, N ⊂ M is

N : L = Ann(N + L/N)

Pro**of**. If a ∈ N : L then aL ⊂ N. Therefore a(N + L) ⊂ N and a ∈ Ann(N +

L/N). Conversely if a ∈ Ann(N + L/N) then aL ⊂ N and therefore a ∈ N :

L.

2.3.18. Exercise. (1) Give an example **of** a homomorphism f : M → N submodules

M1, M2 ⊂ M such that f(M1 ∩ M2) = f(M1) ∩ f(M2).

28 2. MODULES

(2) Give an example **of** a homomorphism f : M → N submodules N1, N2 ⊂ N such

that f −1 (N1 + N2) = f −1 (M1) + f −1 (M2).

(3) Let R be a ring and M a module. Show that M may be regarded as an R/ Ann(M)module

in a natural way.

(4) Let L, N ⊂ M be submodules. Show that Ann(L + N) = Ann(L) ∩ Ann(N).

(5) Let f : M → N be a surjective homomorphism. Show that Ann(M) ⊂ Ann(N).

(6) Let f : M → N be an injective homomorphism. Show that Ann(N) ⊂ Ann(M).

2.4. Sum and product

2.4.1.

Lemma. Let R be a ring and (Mα) a family **of** modules. The product

α Mα is the abelian group **of** all families (xα), xα ∈ Mα with term wise addition.

The setting

r(xα) = (rxα)

is a scalar multiplication on

α Mα. The direct sum

α Mα is the subgroup

**of**

α Mα consisting **of** families with only finitely many nonzero terms. This is a

submodule.

Pro**of**. The laws in 2.1.1 are true for each factor and therefore trivially verified for

the product and sum.

2.4.2. Definition. Let R be a ring and Mα a family **of** modules. By 2.4.1 there are

modules and homomorphisms

(1) The direct product is

α Mα.

(2) The projections pβ :

α Mα → Mβ are the homomorphisms pβ((xα)) =

xβ.

(3) The direct sum is

α Mα. Elements in

α Mα

are written as finite sums

xα.

(4) The injections iβ : Mβ →

α Mα are the homomorphisms given by iβ(x) =

(xα), where xβ = x and xα = 0, α = β.

2.4.3. Proposition. Let R be a ring and Mα a family **of** modules.

(1) Given a family **of** homomorphisms fα : L → Mα, then there exists a unique

homomorphism f : L → Mα such that fα = pα ◦ f.

L

fα

f

Mα

pα

α Mα

(2) Given a family **of** homomorphisms gα : Mα → L, then there exists a unique

homomorphism g : Mα → L such that gα = g ◦ iα.

α Mα

g

iα

gα

Mα

Pro**of**. (1) f(y) = (fα(y)) is the unique homomorphism. (2) g( xα) = gα(xα)

is well defined since only finitely many xα = 0 and a homomorphism.

L

2.4. SUM AND PRODUCT 29

2.4.4. Definition. A family **of** submodules Mα ⊂ M constitutes a direct sum if

any element x ∈

α Mα has a unique finite representation

x =

xα, xα ∈ Mα

2.4.5. Proposition. The following conditions are equivalent

(1) The family Mα ⊂ M constitutes a direct sum.

(2) The natural homomorphism

Mα →

is an isomorphism.

(3) For all β

α

α

α

Mα

Mβ ∩

Mα = 0

α=β

Pro**of**. (1) and (2) are equivalent. If (1) is true and xβ =

α=β xα ∈ Mβ ∩

α=β Mα, then xβ −

α=β xα = 0. Therefore by uniqueness xβ = 0 and (3)

is true. Conversely if (3) is true and

α xα = 0, then xβ = −

α=β xα ∈

Mβ ∩

α=β Mα = 0. for any β. This shows uniqueness and therefore (1) is

true.

2.4.6. Definition. Let R be a ring. A module isomorphic to a direct sum

α

R **of**

copies **of** the ring R is a free module.

A basis **of** a module, is a subset Y such that any element admits a unique finite

representation

α aαyα, where aα ∈ R, yα ∈ Y .

The standard basis **of**

α R consists **of** the elements eα, where each is a family

with 0 for β = α and exactly 1 at index α.

2.4.7. Proposition. A module is free if and only if it admits a basis. If yα is a basis

**of** F then there is an isomorphism

f :

R → F

where f( aα) = aαyα.

α

Pro**of**. Given a free module f :

α R → F then yα = f(eα) is a basis. Conversely

given a basis yα ∈ F then 1yα : R → F 2.3.11 is a family **of** homomorphisms

giving a homomorphism f :

α R → F by 2.4.3. As f( aα) = aαyα

it follows that f is bijective and therefore by 2.1.5 an isomorphism.

2.4.8. Remark. The polynomial ring R[X1, . . . , Xn] is free as R-module.

2.4.9. Example. (1) A nonzero ideal is a free module if and only if it has a basis

consisting **of** a nonzero divisor. Namely if x1 = x2 where in a basis then the

product x1x2 has two different representations.

(2) Let I ⊂ R be an ideal. The module R/I is free if and only if I = 0 or I = R.

2.4.10. Proposition. Any module over a field is free. Conversely if any module

over a nonzero ring is free, then the ring is a field

Pro**of**. By Zorn’s lemma any vector space admits a basis. If I ⊂ R is an ideal and

R/I is free, then I = Ann(R/I) is either 0 or R. So R is a field.

30 2. MODULES

2.4.11. Proposition. Let F be a free module with basis yα. For a module M and

a family **of** elements xα ∈ M there is a unique homomorphism g : F → M such

that g(yα) = xα given by g( aαyα) = aαxα.

Pro**of**. The basis yα ∈ F gives the isomorphism 2.4.7 f :

α R → F . The

family 1xα : R → M gives a homomorphism g ′ :

R → M by 2.4.3. Then

g = g ′ ◦ f −1 .

2.4.12. Corollary. Let M be an R-module and

M R the free module with basis

ex indexed by x ∈ M. The homomorphism

R → M, axex ↦→ ax x

M

is surjective identifying M as a factor module **of** a free module in a natural way.

2.4.13. Definition. A module is indecomposable if it is not isomorphic to a direct

sum **of** two nonzero submodules, otherwise decomposable.

2.4.14. Example. Q is an indecomposable Z-module. Namely if m1

n1

nonzero numbers in two submodules, then n1m2 m1

n1

ber in the intersection.

α

= n2m1 m2

n2

, m2

n2 are

is a nonzero num-

2.4.15. Exercise. (1) Show that if a ring is decomposable as a module, then it is the

product **of** two nonzero rings.

(2) Let Mα be a finite family **of** modules. Show that Mα = Mα,

(3) Let Nα ⊂ Mα be a family **of** submodules modules. Show that

Mα/ Nα Mα/Nα

and that Mα/ Nα Mα/Nα

2.5. Homomorphism modules

2.5.1. Lemma. Let R be a ring and f, g : M → N homomorphisms.

(1) (f + g)(x) = f(x) + g(x) is a homomorphism.

(2) If a ∈ R, then (af)(x) = af(x) is a homomorphism.

Pro**of**. Calculate according to 2.1.1. (1) (f + g)(x + y) = f(x + y) + g(x + y) =

f(x) + f(y) + g(x) + g(y) = (f + g)(x) + (f + g)(y) and (f + g)(ax) =

f(ax) + g(ax) = a(f(x) + g(x)) = a(f + g)(x). (2) (af)(x + y) = af(x + y) =

af(x) + af(y) = (af)(x) + (af)(y) and (af)(bx) = af(bx) = abf(x) =

baf(x) = b(af)(x). The last calculation uses that R is commutative 1.1.2 (4).

2.5.2. Definition. Let R be a ring and M, N modules. By 2.5.1, the homomorphism

module HomR(M, N) is the additive group **of** all homomorphism with

scalar multiplication

R × HomR(M, N) → HomR(M, N), (a, f) ↦→ af = [x ↦→ af(x)]

2.5.3. Definition. Let a ∈ R be a ring and f : M → M ′ , g : N → N ′ , h, k :

M ′ → N homomorphisms **of** modules. By 2.5.1

(1) (h + k) ◦ f = h ◦ f + k ◦ f.

(2) (ah) ◦ f = a(h ◦ f).

(3) g ◦ (h + k) = g ◦ h + g ◦ k.

(4) g ◦ (ah) = a(g ◦ h).

There is induced a homomorphism

**of** R-modules.

2.5. HOMOMORPHISM MODULES 31

Hom(f, g) : HomR(M ′ , N) → HomR(M, N ′ )

(h : M ′ → N) ↦→ (g ◦ h ◦ f : M → N ′ )

2.5.4. Definition. Let R, S be a rings. A functor is a construction T , which to

R-modules M, N associates S-modules T (M), T (N) and a homomorphism

such that

HomR(M, N) → HomS(T (M), T (N)), f ↦→ T (f)

(1) T (1M) = 1 T (M)

(2) T (g ◦ f) = T (g) ◦ T (f)

In case the homomorphism goes

and

HomR(M, N) → HomS(T (N), T (M)), f ↦→ T (f)

(1) T (1M) = 1 T (M)

(2) T (g ◦ f) = T (f) ◦ T (g)

the functor is contravariant. Clearly functors transform isomorphisms into isomorphism.

Given functors T, T ′ a natural homomorphism is a family νM : T (M) → T ′ (M)

**of** homomorphisms, such that for each f : M → N the following diagram commutes

T (M)

T (f)

νN

T (N)

In the contravariant case the diagram is

T (M)

T (f)

T (N)

νM

T ′ (M)

νM

T ′ (f)

T ′ (N)

T ′ (M)

T ′ (f)

νN

T ′ (N)

A natural isomorphism is a natural homomorphism such that each νM is an isomorphism.

2.5.5. Proposition. Let R be a ring.

(1) The construction

is a functor.

(2) The construction

is a contravariant functor

N ↦→ HomR(M, N), g ↦→ Hom(1M, g)

M ↦→ HomR(M, N), f ↦→ Hom(f, 1N)

32 2. MODULES

Pro**of**. By 2.5.3 the construction on homomorphisms are homomorphisms. Given

also homomorphisms f ′ : M ′ → M ′′ and g ′ : N ′ → N ′′ . Then

and

Hom(1M, g ′ ◦ g) = Hom(1M, g ′ ) ◦ Hom(1M, g)

Hom(f ′ ◦ f, 1N) = Hom(f, 1N) ◦ Hom(f ′ , 1N)

showing the conditions on compositions.

2.5.6. Corollary. Let a ∈ R give scalar multiplications aM, aN.

(1)

Hom(aM, 1N) = Hom(1M, aN) : HomR(M, N) → HomR(M, N) f ↦→ af

is scalar multiplication a HomR(M,N).

(2) The map R → HomR(M, M), a ↦→ aM is a homomorphism.

2.5.7. Example. Let R = R1 × R2 be the product ring. The constructions in 2.2.9

is: (1) A functor which to a pair **of** an R1-module and an R2-module associates an

R-module.(2) A functor which to an R-module associates a pair **of** an R1-module

and an R2-module.

2.5.8. Proposition. Let R be a ring and Mα a family **of** modules. For any module

N there are natural isomorphisms

(1) HomR(

α Mα, N)

α HomR(Mα, N)

(2) HomR(N,

α Mα)

α HomR(N, Mα)

Pro**of**. This is 2.4.3 reformulated. (1) A homomorphism g :

α Mα → N is

uniquely determined by the family gα = g ◦ iα : Mα → N. (2) A homomorphism

f : N →

α Mα is uniquely determined by the family fα = pα ◦f : N → Mα.

2.5.9. Lemma. Let R be a ring and M, N modules. For x ∈ M there is a homomorphism

HomR(M, N) → N, f ↦→ f(x).

Pro**of**. Calculate according to 2.1.1 (f + g) ↦→ (f + g)(x) = f(x) + g(x) and

(af) ↦→ (af)(x) = af(x).

2.5.10. Definition. The natural homomorphism 2.5.9

is the evaluation at x.

evx : HomR(M, N) → N, f ↦→ f(x)

2.5.11. Lemma. There is a natural homomorphism

M → HomR(HomR(M, N), N), x ↦→ evx

Pro**of**. Calculate according to 2.1.1 evx+y(f) = f(x + y) = f(x) + f(y) =

evx(f) + evy(f). and evax(f) = f(ax) = af(x) = aevx(f) to see that the map

is a homomorphism.

2.5.12. Proposition. Let R be a ring and M a module. The evaluation

ev1 : HomR(R, M) M, f ↦→ f(1)

is a natural isomorphism. x ↦→ 1x 2.3.11 is the inverse.

Pro**of**. Calculate the composite ev1(x ↦→ 1x) = 1x(1) = x and 1 f(1)(a) =

af(1) = f(a) proving the claims.

2.6. TENSOR PRODUCT MODULES 33

2.5.13. Definition. Let R be a ring and M a module. The dual module is

M ∨ = HomR(M, R)

If f : M → N is a homomorphism, then the dual homomorphism is

f ∨ = Hom(f, 1R) : N ∨ → M ∨

This construction is a contravariant functor.

2.5.14. Lemma. There is a natural homomorphism

where evx(f) = f(x) 2.5.10.

M → M ∨∨ = HomR(HomR(M, R), R), x ↦→ evx

Pro**of**. This is a special case **of** 2.5.11

2.5.15. Definition. A module M is a reflexive module if the homomorphism 2.1.14,

is an isomorphism.

2.5.16. Example. Let R be a ring.

M → M ∨∨

(1) The module R is reflexive.

(2) If (a) = R, (0) in a domain, then R/(a) is not reflexive.

2.5.17. Exercise. (1) Show that HomZ(Q, Z) = 0.

(2) Calculate HomZ(Z/(m), Z/(n)) = 0 for integers m, n.

(3) Let I ⊂ R be an ideal and M a module. Show that HomR(R/I, M) = {x ∈ M|I ⊂

Ann(x)}.

(4) Let R be a ring. Show that a free module with a finite basis is a reflexive module.

(5) If (n) ⊂ (m) ⊂ Z, then show that (m)/(n) is a reflexive Z/(n)-module.

2.6. Tensor product modules

2.6.1. Definition. Let R be a ring and M, N modules. The tensor product module

M ⊗R N = F/F ′

is the factor module F/F ′ , where F = ⊕M×NR is the free module with basis

(x, y) = e (x,y) 2.4.6 and F ′ is the submodule generated by all elements **of** form

(x1 + x2, y) − (x1, y) − (x2, y), (x, y1 + y2) − (x, y1) − (x, y2)

(ax, y) − a(x, y), (x, ay) − a(x, y)

The projection **of** the basis element (x, y) onto M ⊗R N is x ⊗ y = (x, y) + F ′ .

2.6.2. Remark. The relations are interpreted.

(1) There are identities in M ⊗R N

(x1 + x2) ⊗ y = x1 ⊗ y + x2 ⊗ y, x ⊗ (y1 + y2) = x ⊗ y1 + x ⊗ y2

(2) The map

ax ⊗ y = a(x ⊗ y) = x ⊗ ay

⊗ : M × N → M ⊗R N, (x, y) ↦→ x ⊗ y

has partial maps x ↦→ x⊗y : M → M ⊗R N and y ↦→ x⊗y : N → M ⊗R N

that are all homomorphisms.

34 2. MODULES

(3) The formation **of** partial homomorphism are again homomorphisms.

and

N → HomR(M, M ⊗R N)), y ↦→ (x ↦→ x ⊗ y)

M → HomR(N, M ⊗R N)), x ↦→ (y ↦→ x ⊗ y)

2.6.3. Proposition. Given a map µ : M × N → L such that the partial maps x ↦→

µ(x, y) : M → L and y ↦→ µ(x, y) : N → L are homomorphisms. Then there

exists a unique homomorphism u : M ⊗R N → L such that u(x ⊗ y) = µ(x, y).

⊗

M × N

M ⊗R N

µ u

L

Pro**of**. By 2.6.1 M ⊗R N = F/F ′ . The homomorphism 2.4.11 F → K, (x, y) ↦→

µ(x, y) has F ′ in the kernel. 2.3.5 gives the homomorphism u.

2.6.4. Remark. Two homomorphisms u, v : M ⊗RN → L are equal if u(x⊗y) =

v(x ⊗ y) for all x ∈ M, y ∈ N.

2.6.5. Proposition. Let R be a ring and f : M → M ′ , g : N → N ′ homomorphisms

**of** modules. Then there is induced a homomorphism

Pro**of**. The map south-east

M ⊗R N → M ′ ⊗R N ′ , x ⊗ y ↦→ f(x) ⊗ g(y)

M × N

f×g

M ′ × N ′

⊗

M ⊗R N

⊗

M ′

⊗R N ′

satisfies the assumptions in 2.6.3 to induce the right vertical map x ⊗ y ↦→ f(x) ⊗

g(y).

2.6.6. Definition. f ⊗ g : M ⊗R N → M ′ ⊗R N ′ is the induced homomorphism

2.6.5.

2.6.7. Proposition. Let R be a ring. The constructions

(1)

(2)

are functors.

M ↦→ M ⊗R N, f ↦→ f ⊗ 1N

N ↦→ M ⊗R N, g ↦→ 1M ⊗ g

Pro**of**. Given also homomorphisms f ′ : M ′ → M ′′ and g ′ : N ′ → N ′′ . Then by

2.6.4

f ′ ◦ f ⊗ g ′ ◦ g = f ′ ⊗ g ′ ◦ f ⊗ g

The rest follows directly from 2.6.5.

2.6. TENSOR PRODUCT MODULES 35

2.6.8. Corollary. Let a ∈ R give scalar multiplications aM, aN. The homomorphisms

aM ⊗ 1N = 1M ⊗ aN : M ⊗R N → M ⊗R N, x ⊗ y ↦→ a(x ⊗ y)

is scalar multiplication aM⊗RN.

2.6.9. Example. Let R be a ring.

(1) Then there is an isomorphism

R ⊗R R R, a ⊗ b ↦→ ab

(2) Let M, N, L be modules. Composition **of** maps gives a homomorphism

HomR(N, L) ⊗R HomR(M, N) → HomR(M, L), g ⊗ f ↦→ g ◦ f

by 2.5.3. This is a natural homomorphism in each variable.

(3) For a module M composition

HomR(M, M) ⊗R HomR(M, M) → HomR(M, M), g ⊗ f ↦→ g ◦ f

gives HomR(M, M) a structure **of** a normally noncommutative ring. The

map R → HomR(M, M), a ↦→ aM is a ring homomorphism.

2.6.10. Proposition. Let R be a ring and M, N, L modules. Then there are natural

isomorphisms

(1) M ⊗R R M, x ⊗ a ↦→ ax

(2) (M ⊗R N) N ⊗R M, x ⊗ y ↦→ y ⊗ x

(3) (M ⊗R N) ⊗R L M ⊗R (N ⊗R L), (x ⊗ y) ⊗ z ↦→ x ⊗ (y ⊗ z)

Pro**of**. (1) M × R → M, (x, a) ↦→ ax induces the homomorphism M ⊗R R →

M, x ⊗ a ↦→ ax by 2.6.3. The map M → R ⊗R M, x ↦→ 1 ⊗ x is the inverse. (2)

M × N → N ⊗R M, (x, y) ↦→ y ⊗ x induces the homomorphism (M ⊗R N)

N ⊗R M, x ⊗ y ↦→ y ⊗ x by 2.6.3. The inverse is constructed similarly and the

composites are the identities by the uniqueness statement in 2.6.3. (3) For a fixed

z ∈ L the map M × N → M ⊗R (N ⊗R L), (x, y) ↦→ x ⊗ (y ⊗ z) induces the

homomorphism µz : M ⊗R N → M ⊗R (N ⊗R L), x ⊗ y ↦→ x ⊗ (y ⊗ z) by 2.6.3.

Finally the map (M⊗RN)×L → M⊗R(N⊗RL), (x⊗y, z) ↦→ µz(x⊗y). induces

the homomorphism (M ⊗RN)⊗RL M ⊗R(N ⊗RL), (x⊗y)⊗z ↦→ x⊗(y⊗z)

by 2.6.3. The inverse is constructed similarly and the composites are the identities

by the uniqueness statement in 2.6.3.

2.6.11. Proposition. Let R be a ring and Mα a family **of** modules. For any module

N there is a natural isomorphism

(

Mα) ⊗R N

(Mα ⊗R N)

α

giving the identification ( xα) ⊗ y = (xα ⊗ y).

Pro**of**. 2.4.3 and 2.6.3 give a pair **of** inverse homomorphisms. Fix y ∈ N. The

family gα : Mα → (Mα ⊗R N), xα ↦→ xα ⊗ y induces by 2.4.3 a homomorphism

gy : Mα → (Mα ⊗R N). The map ( Mα) × N → (Mα ⊗R

N), ( xα, y) ↦→ gy( xα) induces by 2.6.3 the homomorphism ( Mα) ⊗R

N → (Mα⊗RN), ( xα)⊗y ↦→ xα⊗y. The family iα⊗1N : Mα⊗RN →

( Mα) ⊗R N induces by 2.4.3 the inverse.

α

36 2. MODULES

2.6.12. Example. Let R be a ring, F a free module with basis yα and G a free

module with basis zβ. Then F ⊗R G is a free module with basis yα ⊗ zβ.

2.6.13. Proposition. Let R be a ring and M, N, L modules. Then there is a natural

isomorphism

HomR(M ⊗R N, L) HomR(M, HomR(N, L))

(x ⊗ y ↦→ g(x)(y)) ← g

f ↦→ (x ↦→ [y ↦→ f(x ⊗ y)])

Pro**of**. A given f : M ⊗R N → L is mapped to the composite homomorphism

M → HomR(N, M ⊗R N) → HomR(N, L), 2.5.4 and 2.6.2. This is a homomorphism

as map **of** f by 2.5.4. Given g : M → HomR(N, L) the map M × N →

L, (x, y) ↦→ g(x)(y) induces a homomorphism M ⊗R N → L, x ⊗ y ↦→ g(x)(y)

by 2.6.3. Clearly the maps are inverse to each other and therefore giving an isomorphism

by 2.1.5.

2.6.14. Exercise. (1) Show that Q ⊗Z Q/Z 0.

(2) Show that Z/(m) ⊗Z Z/(n) = 0 if (m, n) = Z.

(3) Let P, Q ⊂ R be different maximal ideals and M a module. Show that M/P M ⊗R

M/QM = 0.

2.7. Change **of** rings

2.7.1. Proposition. (1) Let φ : R → S be a ring homomorphism and N an S

module. The restriction scalars 2.1.11 viewing N as an R-module through

φ, R × N → N, (a, x) ↦→ ax = φ(a)x, is a functor from S-modules to

R-modules.

(2) Let I ⊂ R be an ideal. Restriction **of** scalars along R → R/I identifies

R/I-modules M with R-modules such that I ⊂ Ann(M). For R/I-modules

M, N there is a natural isomorphism

HomR(M, N) Hom R/I(M, N)

Pro**of**. This is a restatement **of** 2.1.11 using 2.5.4.

2.7.2. Lemma. Let R → S be a ring homomorphism, M an R-module and N an

S-module. Then

is an S-scalar multiplication.

S × M ⊗R N → M ⊗R N, (b, x ⊗ y) ↦→ x ⊗ by

Pro**of**. For fixed b ∈ S the map M × N → M ⊗R N, (x, y) ↦→ x ⊗ by induces

the homomorphism µb : M ⊗R N → M ⊗R N, x ⊗ y ↦→ x ⊗ by by 2.6.3. This

gives a well defined scalar multiplication S × M ⊗R N → M ⊗R N, (b, x ⊗ y) ↦→

µb(x ⊗ y).

2.7.3. Definition. Let R → S be a ring homomorphism and M an R-module. The

change **of** ring S-module is M ⊗R S with S-scalar multiplication 2.7.2

S × M ⊗R S → M ⊗R S, (b, x ⊗ c) ↦→ x ⊗ bc

2.7.4. Proposition. The construction

2.7. CHANGE OF RINGS 37

M ↦→ M ⊗R S

and

f : M → M ′ ↦→ f ⊗ 1S : M ⊗R S → M ′ ⊗R S

is a functor from R-modules to S-modules.

Pro**of**. This is clear from 2.7.2 and 2.6.7.

2.7.5. Proposition. Let R → S be a ring homomorphism, M an R-module and N

an S-modules. Then there is a natural isomorphism **of** S-modules.

M ⊗R S ⊗S N M ⊗R N, x ⊗ b ⊗ y ↦→ x ⊗ by

Pro**of**. The homomorphism v : S ⊗S N → N, b ⊗ y ↦→ by is an isomorphism,

2.6.10. The homomorphism 1M ⊗ v : M ⊗R S ⊗S N M ⊗R N is an R-module

isomorphism, 2.6.7. The identity x ⊗ bc ⊗ y = x ⊗ b ⊗ cy proves this to be an

S-module homomorphism.

2.7.6. Proposition. Let R → S be a ring homomorphism, M an R-module and N

an S-modules. Then there is a natural isomorphism

HomR(M, N) HomS(M ⊗R S, N), f ↦→ (x ⊗ b ↦→ bf(x))

Pro**of**. A given f is mapped to the composite M⊗R → N ⊗R S → N which is an

S-homomorphism. Given a homomorphism g : M ⊗R S → N then the composite

M → M ⊗R S → N is an R-homomorphism and an inverse to the first given

map.

2.7.7. Lemma. Let R → S be a ring homomorphism, M an R-module and N an

S-module. Then

S × HomR(N, M) → HomR(N, M), (b, f : N → M) ↦→ (y ↦→ f(by))

is an S-scalar multiplication.

Pro**of**. The map (b, f) ↦→ f ◦ bN satisfies the laws 2.1.1.

2.7.8. Definition. Let R → S be a ring homomorphism and M an R-module. The

induced module is the S-module HomR(S, M) with S-scalar multiplication 2.7.7

S × HomR(S, M) → HomR(S, M), (b, f : S → M) ↦→ (c ↦→ f(bc))

2.7.9. Proposition. The induced module

M ↦→ HomR(S, M)

and

f : M → M ′ ↦→ Hom(1S, f) : HomR(S, M) → HomR(S, M ′ )

is a functor from R-modules to S-modules.

Pro**of**. This is clear from 2.7.7 using 2.5.4.

2.7.10. Proposition. Let R → S be a ring homomorphism and M an R-module

and N an S-modules. Then there is a natural isomorphism

HomR(N, M) HomS(N, HomR(S, M)), f ↦→ (y ↦→ [b ↦→ f(by)])

Pro**of**. g ↦→ (y ↦→ g(y)(1)) is an inverse.

2.7.11. Example. Let I ⊂ R be an ideal and R → R/I the projection.

38 2. MODULES

(1) The change **of** ring functor maps an R-module M to the R/I-module M/IM.

The natural isomorphism 2.7.6 is

HomR(M, N) Hom R/I(M/IM, N)

for any R/I-module N.

(2) The induced module functor maps an R-module M to the R/I-module {x ∈

M|I ⊂ Ann(x)}. The natural isomorphism 2.7.10 is

HomR(N, M) Hom R/I(N, HomR(R/I, M))

for any R/I-module N.

2.7.12. Definition. Let R → S, S ′ be ring homomorphisms. The tensor product

ring over R is S ⊗R S ′ with multiplication given by (b ⊗ b ′ )(c ⊗ c ′ ) = bc ⊗ b ′ c ′

extended by linearity. R → S ⊗R S ′ , r ↦→ r ⊗ 1 = 1 ⊗ r is the natural ring

homomorphism.

2.7.13. Proposition. Let φ, φ ′ : R → S, S ′ and ψ, ψ ′ : S, S ′ → T give a commutative

diagram **of** ring homomorphisms, ψ ◦ φ = ψ ′ ◦ φ ′ . Then b ⊗ b ′ ↦→ ψ(b)ψ ′ (b ′ )

is the unique homomorphism making the following diagram commutative.

Pro**of**. This is clear by 2.6.3.

R

S ′

S

S ⊗R S

′

T

2.7.14. Example. Let R → S be a ring homomorphism. Then

is an isomorphism.

R[X] ⊗R S S[X]

2.7.15. Exercise. (1) Show that the change **of** rings **of** a free R-module is a free Smodule.

(2) Let φ : R → S be a ring homomorphism. Show that the change **of** rings **of** a

scalar multiplication a : M → M on an R-module is a scalar multiplication φ(a) :

M ⊗R S → M ⊗R S.

(3) Show that the change **of** rings **of** the composition **of** two homomorphisms is the

composition **of** the change **of** rings **of** each homomorphism.

(4) Show the isomorphism

R[X] ⊗R R[Y ] R[X, Y ]

3

Exact sequences **of** modules

3.1. Exact sequences

3.1.1. Definition. Let f : M → N and g : N → L be homomorphisms **of**

modules. The sequence

**of** homomorphisms is a

M f

g

N

L

(1) 0-sequence: g ◦ f = 0 or equivalently Im f ⊂ Ker g

(2) exact sequence: Im f = Ker g

For a sequence **of** more homomorphisms the conditions should be satisfied for

every consecutive composition. E.g. The sequence

M f

g

N

h

L

K

is a 0-sequence if g ◦ f = 0 and h ◦ g = 0. The sequence is exact if Im f = Ker g

and Im g = Ker h.

3.1.2. Remark. An interpretation **of** 2.3.3 gives:

(1) The sequence

is exact if and only if f is injective.

(2) The sequence

0

M f

f

M

N

is exact if and only if f is surjective.

(3) The sequence

0

f

M

N

is exact if and only if f is an isomorphism.

3.1.3. Proposition. (1) For a homomorphism f : M → N the sequence

0

Ker f

f

M

N

N

0

0

Cok f

is exact.

(2) For scalar multiplication with a ∈ R on M the sequence

0

is exact.

Ker aM

aM

M

M

39

M/aM

0

0

40 3. EXACT SEQUENCES OF MODULES

3.1.4. Proposition. The 0-sequence

0

f

M

g

N

is exact if and only if the following equivalent statements are satisfied.

(1) f is an isomorphism onto Ker g.

(2) Given a homomorphism h : K → N such that g ◦ h = 0 then there is a

unique h ′ : K → M such that h = f ◦ h ′ .

0

f g

M

N

h

K

′

h

Pro**of**. (1) This is clearly equivalent with exactness. (2) Assume the sequence exact.

Im h ⊂ Ker g = Im f, so by (1) put h ′ = f −1 ◦ h. Assume (2) satisfied and

apply it to Ker g → M to see that (1) is satisfied.

3.1.5. Proposition. The 0-sequence

M f

g

N

is exact if and only if the following equivalent statements are satisfied.

(1) The factor homomorphism 2.3.9 g ′ : Cok f → L induced by g is an isomorphism.

(2) Given a homomorphism k : N → K such that k ◦ f = 0 then there is a

unique k ′ : L → K such that k = k ′ ◦ g.

M f

L

g

N

L

k

k

′

K

Pro**of**. (1) The equivalence follows from 2.3.9. (2) Assume the sequence exact. By

2.3.5 there is k ′′ : Cok f → K such that k ′′ ◦ p = k. By (1) put k ′ = k ′′ ◦ g −1 .

Assume (2) satisfied and apply it to N → Cok f to see that (1) is satisfied.

3.1.6. Proposition. Let

Mα

Nα

Lα

L

L

be a family **of** exact sequences. Then there are exact sequences:

(1) The sum

Mα

Nα

Lα

(2) The product

Mα

Nα

0

0

Lα

Pro**of**. Clear since kernel and image is calculated componentwise.

3.1.7. Definition. An exact sequence

0

f

M

g

N

is a short exact sequence. That is f is injective, Im f = Ker g and g is surjective.

L

0

3.1. EXACT SEQUENCES 41

3.1.8. Proposition.

quence

(1) Let I ⊂ R be an ideal, then there is a short exact se-

0 I R R/I 0

(2) Let M ⊂ N be a submodule, then there is a short exact sequence

0

M

N

N/M

(3) For scalar multiplication with nonzero divisor a ∈ R on M the sequence

0

aM

M

M

M/aM

is a short exact sequence.

(4) Given a homomorphism f : M → N there are associated two short exact

sequences.

and

0

0

Ker f

Im f

f

M

Im f

N

Cok f

(5) For scalar multiplication with any a ∈ R on M there are associated two short

exact sequences.

and

0

0

Ker aM

aM

M

aM

M

aM

M/aM

3.1.9. Definition. Let f : M → N be a homomorphism.

(1) f has a retraction if there is a homomorphism u : N → M such that u ◦ f =

1M.

(2) f has a section if there is a homomorphism v : N → M such that f ◦v = 1N.

3.1.10. Proposition. Let f : M → N be a homomorphism.

(1) If f has a retraction u : N → M then f is injective, u is surjective and

N = Im f ⊕ Ker u

(2) If f has a section v : N → M then f is surjective, v is injective and

M = Ker f ⊕ Im v

Pro**of**. (1) u(f(x)) = x so f is injective and u is surjective. If y ∈ N then

y = f(u(y)) + (y − f(u(y)) and u(y − f(u(y))) = 0, so N = Im f + Ker u. Let

y ∈ Im f ∩ Ker u. Then y = f(x) gives x = u(f(x)) = u(y) = 0, so y = 0.

Conclude by 2.4.5 that the sum is direct. (2) y = f(v(y)) so f is a retraction **of** v.

Finish by (1).

3.1.11. Lemma. For a short exact sequence

the following are equivalent

0

(1) f has a retraction.

(2) g has a section.

f

M

g

N

L

0

0

0

0

0

0

0

42 3. EXACT SEQUENCES OF MODULES

For any retraction u there is a unique section v and wise-verse such that

1N = f ◦ u + v ◦ g

Pro**of**. If u is a retraction **of** f, then Ker g = Im f ⊂ Ker(1N − f ◦ u). By 3.1.5

there is a homomorphism v : L → N such that v ◦ g = 1N − f ◦ u. This is a

section **of** g. Conversely if v is a section **of** g then Im(1N − v ◦ g) ⊂ Ker g, so

there is a homomorphism u : N → M such that f ◦ u = 1N − v ◦ g, 3.1.4. u is a

retraction **of** f. The equation is clearly satisfied.

3.1.12. Definition. Let R be a ring and f : M → N, g : N → L homomorphisms.

A short exact sequence

0

f

M

g

N

is a split exact sequence if equivalently 3.1.11 f has a retraction or g has a section.

3.1.13. Proposition. A sequence

0

f

M

g

N

is a split exact sequence if and only if there are homomorphism u : N → M, v :

L → N satisfying

L

L

0

0

g ◦ f = 0, u ◦ f = 1M, g ◦ v = 1L, f ◦ u + v ◦ g = 1N

If the sequence is split exact then

0

v

L

u

N

N

is split exact and (x, y) ↦→ f(x)+v(y) and z ↦→ u(z)+g(z) gives the isomorphism

M ⊕ L N

Pro**of**. The sequence is a 0-sequence f is injective and g is surjective. From f ◦

u + v ◦ g = 1N follows that z ∈ Ker g ⊂ Im f, so the sequence is short exact. The

rest is contained in 3.1.10.

3.1.14. Corollary. A (contravariant) functor preserves split exact sequences. If

0

f

M

is split exact and T a functor, then

is split exact.

0

g

N

L

T (M) T (f)

T (N) T (g)

T (L)

Pro**of**. By 3.1.13 a split exact sequence is characterized by a set **of** equations.

These are preserved by the functor, 2.5.4.

3.1.15. Example. A short exact sequence

0

f

M

g

N

where L is a free module is a split exact sequence. Namely let xα ∈ L be a

basis and choose yα ∈ N with g(yα) = xα. The define v : L → N by setting

v(xα) = yα, 2.4.11.

L

0

0

0

0

3.2. THE SNAKE LEMMA 43

3.1.16. Example. Let Zi is the family **of** modules each a copy **of** Z indexed by the

natural numbers. Then the short exact sequence

0

i Zi

i Zi

is not split exact.

The element f = (1, 2, 22 , . . . , 2n , . . . ) +

fk = (0, . . . , 0, 2 n−k , . . . ) +

But in

i Zi/ i Zi

0

i Zi is divisible by 2 k for all k. If

i Zi for n ≥ k, then 2 k fk = f in

i Zi the only element divisible with all 2 k is 0, so no section exists.

3.1.17. Exercise. (1) Show that the sequence

0

Z

is short exact, but not split exact.

(2) Show that the sequence

0

Q

n

Z

Z

is exact, but not split exact for n = 0, 1.

(3) Show that the sequence

is exact, but not split exact.

(4) Show that the sequence

is split exact.

0

0

Z/(2) 1↦→2

Z/(4)

Z/(2) 1↦→3

Z/(6)

Q/Z

Z/(n)

3.2. The snake lemma

Z/(2)

Z/(3)

3.2.1. Example. Given a commutative diagram **of** homomorphisms

M

u

M ′

there is induced a commutative diagram

0

0

Ker f

u

Ker f ′

M

u

M ′

where the rows are exact sequences.

The diagram splits into two diagrams

0

0

Ker f

u

Ker f ′

f

f ′

f

f ′

M

u

M ′

N

v

N ′

N

0

0

Cok f

0

0

v

v

N ′ Cok f ′

f

Im f

f

v

′

Im f ′

0

0

i

0

0

Zi/

i Zi.

44 3. EXACT SEQUENCES OF MODULES

and

0

0

Im f

N

where the rows are short exact sequences.

Cok f

v

v

v

Im f ′

N ′ Cok f ′

3.2.2. Lemma. Given a commutative diagram **of** homomorphisms

0

M

u

M ′

f

f ′

N

v

N ′

g

g ′

L

w

L ′

where the rows exact sequences. The snake homomorphism δ : Ker w → Cok u

is well defined by: For z ∈ Ker w choose y ∈ N such that g(y) = z. The

element v(y) ∈ Ker g ′ so there is x ′ ∈ M ′ such that f ′ (x ′ ) = v(y). Then δ(z) =

x ′ + Im u ∈ Cok u.

Pro**of**. Assume g(y ′ ) = z and f ′ (x ′′ ) = v(y ′ ). There is x ∈ M with f(x) = y−y ′ .

Now f ′ (u(x)) = v(f(x)) = v(y − y ′ ) = f ′ (x ′ − x ′′ ) so u(x) = x ′ − x ′′ since f ′

is injective. Then x ′ + Im u = x ′′ + Im u as wanted. The choices made respect

addition and scalar multiplication showing that δ is a homomorphism.

3.2.3. Remark. The snake is

Ker u

f

Ker v

0

0

f

g

M N L 0

u

v

w

0

M ′

f ′

N ′

g ′

L ′

Cok u

The construction **of** δ is schematically

M ′

Cok u

f ′

N

v

N ′

g

f ′

Ker w

L

Cok v

g

g ′

x ′

δ(z)

Ker w

Cok w

0

v(y)

z

y

z

3.2. THE SNAKE LEMMA 45

3.2.4. Theorem (snake lemma). Given a commutative diagram **of** homomorphisms

0

M

u

M ′

f

f ′

N

v

N ′

g

g ′

L

w

L ′

where the rows exact sequences. There is induced a six term long exact sequence

Ker u

f

Ker v

δ

f ′

Cok v

Cok u

g

g ′

Ker w

Cok w

Pro**of**. By construction **of** δ it is clear that the sequence is a 0-sequence: If y ∈

Ker v then to calculate δ(g(y)) the choice v(y) = 0 gives δ ◦ g = 0. Also

f ′ (δ(z)) = v(y) + Im v shows that f ′ ◦ δ = 0. Exactness at Ker v and Cok v

are clear. Given z ∈ Ker w such that δ(z) = 0. By 3.2.2 choose y, g(y) = z

and x ′ , f ′ (x ′ ) = v(y). Then δ(z) = x ′ + Im u = 0, so choose x, u(x) = x ′ .

Now v(f(x)) = f ′ (u(x)) = v(y) so y − f(x) ∈ Ker v and g(y − f(x)) =

g(y) = z. Therefore exactness at Ker w. Given x ′ + Im u ∈ Cok u such that

f ′ (x ′ ) + Im v = 0 ∈ Cok v. Choose y, v(y) = f ′ (x ′ ) and put z = g(y). Then

w(g(y)) = g ′ (v(y)) = g ′ (f ′ (x ′ )) = 0. Now z ∈ Ker w and δ(z) = x ′ + Im u.

Therefore exactness at Cok u.

3.2.5. Corollary. If f is injective then the f : Ker u → Ker v is injective and the

long exact sequence is

0

Ker u

f

Ker v

δ

f ′

Cok v

Cok u

g

g ′

0

Ker w

Cok w

If g ′ is surjective then g ′ : Cok v → Cok w is surjective and the long exact sequence

is

Ker u

f

Ker v

δ

f ′

Cok v

Cok u

g

g ′

Ker w

Cok w

3.2.6. Corollary. (1) If v is injective and u is surjective, then w is injective.

(2) If v is surjective and w is injective, than u is surjective.

(3) If v is an isomorphism, then w is injective if and only if u is surjective.

3.2.7. Proposition. Given submodules N, L ⊂ M, then there is a short exact

sequence

0

0

M/N ∩ L x↦→(x,x)

M/N ⊕ M/L

(x,y)↦→x−y

M/N + L

0

46 3. EXACT SEQUENCES OF MODULES

Pro**of**. The commutative diagram

0

0

N ∩ L

N ⊕ L

N + L

x↦→(x,x)

(x,y)↦→x−y

M

M ⊕ M

M

where the rows are short exact sequences, gives by 3.2.4 a five term long exact

sequence

0

M/N ∩ L

M/N ⊕ M/L

M/N + L

3.2.8. Proposition (five lemma). Given a commutative diagram **of** homomorphisms

M1

u1

M ′ 1

M2

u2

M ′

2

M3

u3

M ′

3

M4

u4

M ′

4

M5

u5

M ′

5

where the rows are exact sequences. If u1 is surjective, u2, u4 are isomorphism

and u5 is injective, then u3 is an isomorphism.

Pro**of**. Let fi : Mi → Mi+1, f ′ i : M ′ i → M ′ i+1 . Split the given diagram in three as

follows

0

0

0

0

0

M1

u1

Ker f ′

2

Cok f2

u ′′

3

Ker f ′

4

Im f2

u ′ 3

Im f ′

2

M2

u2

M ′

2

M4

u4

M ′

4

M3

u3

M ′

3

Cok f1

u ′ 3

Im f ′

2

Im f4

u5

M ′

5

Cok f2

u ′′

3

Cok f ′

2

Note that Cok f1 Im f2 and Cok f ′ 2 Ker f ′ 4 . Now use 2.3.3 and the snake

lemma to conclude that Ker u3 = 0 and Cok u3 = 0 and u3 is therefore an isomorphism.

3.2.9. Proposition (windmill lemma). Given homomorphism M f

There is induced an eight term long exact sequence

0

Ker f

Ker g ◦ f

Cok f

f

g

Ker g

Cok g ◦ f

0

0

0

0

0

0

0

Cok g

0

g

N

0

L .

Pro**of**. Look at the two diagrams

0

0

3.2. THE SNAKE LEMMA 47

0

Ker g

M

f

g◦f

L

1

By the snake lemma the sequences

0

N

M

g

L

Ker f

δ=f

Cok f

Ker g

Ker g ◦ f

f

g

N

Ker g

δ=g

Cok g

Cok g ◦ f

1

M

g◦f

L

Cok f

0

are exact and overlap to give the windmill sequence.

3.2.10. Remark. The windmill is

Ker g ◦ f

Cok g ◦ f

Cok f

Ker g ◦ f

Ker g

f

Ker f

M

N

g◦f

g

Cok f

0

L

Cok g

Cok g ◦ f

3.2.11. Exercise. (1) Given a short exact sequence

0

M

N

Show that Ann(N) ⊂ Ann(M) ∩ Ann(L).

(2) Give a short exact sequence

0

M

N

such that Ann(N) = Ann(M) ∩ Ann(L).

(3) Given ideals I, J ⊂ R. Show that there is a short exact sequence.

0

R/I ∩ J

R/I ⊕ R/J

L

L

0

0

0

0

0

R/I + J

0

48 3. EXACT SEQUENCES OF MODULES

3.3. Exactness **of** Hom

3.3.1. Proposition. Given an exact sequence

0

f

M

g

N

L

and an R-module K. Then the following sequence is exact

0

HomR(K, M)

HomR(K, N)

HomR(K, L)

Pro**of**. Given h : K → M such that f ◦ h = 0 then h = 0 since f is injective.

Given k : K → N such that g ◦ k = 0 then by 3.1.4 there is h : K → M such that

f ◦ h = k. So the sequence is exact.

3.3.2. Proposition. Given a sequence

0

f

M

g

N

L

Such that for any module K, the following sequence is exact

0

HomR(K, M)

Then the original sequence is exact.

HomR(K, N)

HomR(K, L)

Pro**of**. For K = M the identity 1M is mapped to g ◦ f ◦ 1M = 0 so it is a 0sequence.

Assume f(x) = 0. Take K = R then 1x is mapped to f ◦ 1x = 1 f(x) =

0. Therefore 1x = 0 and so x = 0. That insures that f is injective. Assume

g(y) = 0. Take K = R then 1y is mapped to g ◦ 1y = 1 g(y) = 0. There exists a

homomorphism h : R → M such that f ◦ h = 1y. By 2.5.12 h = 1x and therefore

f(x) = y. The original sequence is now proven exact.

3.3.3. Proposition. Given an exact sequence

M f

g

N

L

and a module K. Then the following sequence is exact

0

HomR(L, K)

HomR(N, K)

0

HomR(M, K)

Pro**of**. Given h : L → K such that h ◦ g = 0 then h = 0 since g is surjective.

Given k : N → K such that k ◦ f = 0 then by 3.1.5 there is h : L → K such that

h ◦ g = k. So the sequence is exact.

3.3.4. Proposition. Given a sequence

M f

g

N

L

such that for any module K, the following sequence is exact

0

HomR(L, K)

Then the original sequence is exact.

HomR(N, K)

0

HomR(M, K)

Pro**of**. For K = L the identity 1L is mapped to 1L ◦g ◦f = 0 so it is a 0-sequence.

Take K = Cok g then pg : L → Cok g has pg ◦ g = 0, but by exactness 0 is the

unique homomorphism satisfying this, so pg = 0. Therefore Cok g = 0 and g is

surjective. Take K = Cok f, p : N → Cok f the projection. p ◦ f = 0 so by

exactness there exists a unique q : L → Cok f such that q ◦ g = p. It follows that

Ker g ⊂ Ker p = Im f. All together the original sequence is exact.

3.3.5. Proposition. Given a sequence

The following are equivalent.

0

3.4. EXACTNESS OF TENSOR 49

f

M

g

N

(1) The sequence is split exact.

(2) For any K the following sequence is exact

0

HomR(K, M)

HomR(K, N)

(3) For any K the following sequence is exact

0

HomR(L, K)

HomR(N, K)

L

0

HomR(K, L)

HomR(M, K)

If the conditions are true, then the sequences (2) and (3) are split exact.

Pro**of**. (1) ⇒ (2), (1) ⇒ (3) are clear by 3.1.14 giving that the sequences (2), (3)

are split exact. (2) ⇒ (1): Let K = L, then there is a section to g. By 3.3.2 and

3.1.11 the original sequence is split exact. (3) ⇒ (1): Let K = M, then there is a

retraction to f. By 3.3.4 and 3.1.11 the original sequence is split exact.

3.3.6. Exercise. (1) Show that the sequence

0

HomZ(Q/Z, Z)

HomZ(Q/Z, Q)

is exact, but the rightmost map is not surjective.

(2) Show that the sequence

0

HomZ(Z/(n), Z)

n

HomZ(Z/(n), Z)

is exact, but the rightmost map is not surjective.

3.4. Exactness **of** Tensor

3.4.1. Proposition. Given an exact sequence

M f

g

N

L

0

and an R-module K. Then the following sequence is exact

M ⊗R K

N ⊗R K

L ⊗R K

0

0

HomZ(Q/Z, Q/Z)

HomZ(Z/(n), Z/(n))

Pro**of**. Let K ′ be any module. By 3.3.4 it is enough to see that the sequence

0

HomR(L ⊗R K, K ′ )

is exact. By 2.6.13 it amounts to see that the sequence

0

HomR(L, HomR(K, K ′ ))

is exact. This follows from 3.3.3.

0

HomR(N ⊗R K, K ′ )

HomR(M ⊗R K, K ′ )

HomR(M, HomR(K, K ′ ))

HomR(N, HomR(K, K ′ ))

50 3. EXACT SEQUENCES OF MODULES

3.4.2. Proposition. Given a split exact sequence

0

f

M

g

N

L

and a module K. Then the following sequence is split exact

0

K ⊗R M

K ⊗R N

Pro**of**. This follows from the functor properties 3.1.14.

0

K ⊗R L

3.4.3. Proposition. Let I ⊂ R be an ideal. For any module M, the homomorphism

is an isomorphism.

M ⊗R R/I → M/IM, x ⊗ a + I ↦→ ax + IM

Pro**of**. x + IM ↦→ x ⊗ 1 is an inverse.

3.4.4. Corollary. Let I, J ⊂ R be ideals. Then

is an isomorphism.

R/I ⊗R R/J → R/(I + J), a + I ⊗ b + J ↦→ ab + I + J

3.4.5. Proposition. Let I1, . . . , Ik ⊂ R be pairwise comaximal ideals and M a

module. Then the product **of** projections

is an isomorphism.

Pro**of**. By Chinese remainders 1.4.2

M/I1 · · · IkM → M/I1M × · · · × M/IkM

R/I1 · · · Ik → R/I1 × · · · × R/Ik

is an isomorphism. Tensor with M and use 3.4.3 to get the isomorphism.

3.4.6. Exercise. (1) Calculate Z/(m) ⊗Z Z/(n) for all integers m, n.

(2) Let I ⊂ R be an ideal. Show that R/I ⊗R R/I R/I.

(3) Let I ⊂ R be an ideal. Show that I ⊗R R/I I/I 2 .

(4) Let 2Z be scalar multiplication. Show that 2Z ⊗ 1 Z/(2) : Z ⊗Z Z/(2) → Z ⊗Z Z/(2)

is not injective.

3.5. Projective modules

3.5.1. Definition. An R-module F is a projective module if for any exact sequence

N → L → 0 the sequence

is exact.

HomR(F, N) → HomR(F, L) → 0

3.5.2. Proposition. A module F is projective if and only if any surjective homomorphism

M → F → 0 has a section.

Pro**of**. Assume F projective and g : M → F surjective. Then HomR(F, M) →

HomR(F, F ) → 0 is exact. So there exists a v : F → M such that g ◦ v = 1F .

v is then a section. Conversely given g : N → L surjective and h : F → L. Let

M = Ker N ⊕F → L, (y, z) ↦→ g(y)−h(z) and pN : M → N, pF : M → F the

0

3.5. PROJECTIVE MODULES 51

projections, then g ◦ pN = h ◦ pF . Now pF is surjective since g is. Let v : F → M

be a section **of** pF , then h ′ = pN ◦ v satisfies h = g ◦ h ′ .

pN

N g

L

h

M

F

′

h

v

3.5.3. Corollary. A short exact sequence

0

f

M

pF

g

N

0

F

where F is a projective module is a split exact sequence.

3.5.4. Corollary. A free module is projective. Over a field every module is projective.

3.5.5. Example. Let I ⊂ R be an ideal. If R/I is projective, then there is a ring

decomposition R/I × R ′ R.

3.5.6. Proposition. A direct summand in a projective module is projective.

Pro**of**. Let F ⊕ F ′ be projective and g : M → F surjective. By 3.5.2 there is a

section v ′ : F ⊕ F ′ → M ⊕ F ′ to (g, 1F ′). Then v(y) = pM ◦ v ′ (y, 0) is a section

to g and F is projective.

3.5.7. Proposition. A module is projective if and only if it is a direct summand in

a free module.

Pro**of**. By 2.4.12 any module is a factor module **of** a free module. By 3.5.2 a

projective factor module has a section, and is therefore by 3.1.10 a direct summand.

3.5.8. Proposition. Let Fα be a family **of** projective modules, then the direct sum

α Fα is a projective module.

Pro**of**. Let N → L be surjective. Then by 2.5.8

is the product

0

HomR( Fα, N) → HomR( Fα, L)

HomR(Fα, N) → HomR(Fα, L)

which is surjective by 3.1.6. So Fα is projective.

3.5.9. Proposition. Let F, F ′ be projective modules. Then F ⊗R F ′ is projective.

Pro**of**. F ⊗R F ′ is clearly a direct summand in a free module.

3.5.10. Proposition. Let R → S be a ring homomorphism and F a projective

module. The change **of** ring module F ⊗R S is a projective S-module.

Pro**of**. A direct summand **of** a free R-module is changed to a direct summand **of** a

free S-module.

52 3. EXACT SEQUENCES OF MODULES

3.5.11. Proposition. Any module M admits an exact sequence

F → M → 0

where F is a projective module. That is, any module is a factor module **of** a projective

module.

Pro**of**. Take F free, 2.4.12.

3.5.12. Exercise. (1) Let R = R1 × R2. Show that R1, R2 are projective ideals in R.

(2) Show that the ideal (2)/(6) in the ring Z/(6) is projective.

(3) Show that the ideal (2)/(4) in the ring Z/(4) is not projective.

3.6. Injective modules

3.6.1. Definition. An R-module E is an injective module if for any exact sequence

0 → M → N the sequence

is exact.

HomR(N, E) → HomR(M, E) → 0

3.6.2. Proposition. A module E is an injective module if and only if any injective

homomorphism 0 → E → L has a retraction.

Pro**of**. Assume E injective and f : E → L injective. Then HomR(L, E) →

HomR(E, E) → 0 is exact. So there exists a u : L → E such that u ◦ f = 1E.

Then u is a retraction. Conversely given f : M → N injective and h : M → E.

Let L = Cok M → E ⊕ N, x ↦→ h(x) − f(x) and iE : E → L, iN : N → L the

injections, then iN ◦ f = iE ◦ h. Now iE is injective since f is. Let u : L → E be

a retraction **of** iE, then h ′ = u ◦ iN satisfies h = h ′ ◦ f.

0

M

3.6.3. Corollary. A short exact sequence

0

f

E

f

N

h

h

′

E iE

L

u

g

N

iN

L

where E is an injective module is a split exact sequence.

3.6.4. Example. Let I ⊂ R be an ideal. If I is injective, then there is a ring

decomposition R/I × R ′ R.

3.6.5. Proposition. A direct summand in an injective module is injective.

Pro**of**. Let E ⊕ E ′ be an injective module and f : E → L an injective homomorphism.

By 3.6.2 there is a retraction u ′ : L ⊕ E ′ → E ⊕ E ′ to (f, 1E ′). Then

u(y) = pE ◦ u ′ (y, 0) is a retraction to f and E is injective.

3.6.6. Proposition. Let Eα be a family **of** injective modules, then the direct product

α Eα is an injective module

0

Pro**of**. Let M → N be injective. Then by 2.5.8

is the product

3.6. INJECTIVE MODULES 53

HomR(N, Eα) → HomR(M, Eα)

HomR(N, Eα) → HomR(M, Eα)

which is surjective by 3.1.6. So Eα is injective.

3.6.7. Proposition. A module E is injective, if for any ideal I ⊂ R

is exact.

HomR(R, E) → HomR(I, E) → 0

Pro**of**. Let f : E → L be an injective homomorphism. The set **of** submodules

f(E) ⊂ L ′ ⊂ L and retractions u ′ : L ′ → E is nonempty and inductively ordered.

By Zorn’s lemma a maximal L ′ , u ′ exists. If L/L ′ = 0 choose a y ∈ L\L ′ . The

homomorphism Ann(y + L ′ ) → E, a ↦→ u ′ (ay) extends to u ′′ : R → E by

hypothesis. The setting L ′ + Ry → E, x + ay ↦→ u ′ (x) + u ′′ (a) is a well defined

retraction. This contradicts maximality.

3.6.8. Definition. Let R be a domain. M is a divisible module if scalar multiplication

with a nonzero a ∈ R is surjective.

3.6.9. Proposition. (1) An injective module over a domain is divisible.

(2) A divisible module over a principal ideal domain is injective.

(3) Over a field any module is injective.

Pro**of**. (1) Let E be an injective module and a = 0. Choose x ∈ E and look

at 1x : R → E. Scalar multiplication aR is injective, so there is an extension

1y : R → E such that 1x = 1y ◦ aR. Then ay = x. (2) Let E be divisible. To

extend f : (a) → E, choose x ∈ E such that ax = f(a), then 1x : R → E, 1 → x

extends f. (3) This is clear.

3.6.10. Proposition. Let R → S be a ring homomorphism and E an injective

R-module. The induced module HomR(S, E) is an injective S-module.

Pro**of**. Let M → N be an injective homomorphism **of** S-modules. Then by

2.7.10 HomS(N, HomR(S, E)) → HomS(M, HomR(S, E)) is HomR(N, E) →

HomR(M, E) being surjective since E is an injective R-module.

3.6.11. Lemma. (1) Q and Q/Z are divisible and therefore injective Z modules.

(2) The homomorphism 2.5.11

is injective.

x ↦→ evx : M → HomZ(HomZ(M, Q/Z), Q/Z)

Pro**of**. (1) Clear by 3.6.9. (2) Let x = 0 and choose h : Z/ Ann(x) → Q/Z

nonzero. Now Z/ Ann(x) Zx ⊂ M so extend h to h ′ : M → Q/Z. Then

evx(h ′ ) = h(1) = 0.

3.6.12. Proposition. Any module M admits an exact sequence

0 → M → E

where E is an injective module. That is, any module is a submodule **of** an injective

module.

54 3. EXACT SEQUENCES OF MODULES

Pro**of**. By 2.4.12 choose a surjection F → HomZ(M, Q/Z) where F is a free

R-module. Then

0 → M → HomZ(F, Q/Z)

is exact by 3.6.11. Since F

α R the module

HomZ(F, Q/Z)

HomZ(R, Q/Z)

is injective, 3.6.6 and 3.6.10.

3.6.13. Proposition. A homomorphism f : M → N is injective if and only if

is surjective for any injective module E.

HomR(N, E) → HomR(M, E) → 0

Pro**of**. Assume that Ker f = 0 and choose 0 → Ker f → E, 3.6.12. The sequence

HomR(N, E) → HomR(M, E) → HomR(Ker f, E) → 0

is exact. So HomR(N, E) → HomR(M, E) is not surjective.

3.6.14. Definition. A submodule N ⊂ M is an essential extension if any nonzero

submodule L ⊂ M has nonzero intersection N ∩ L = 0. An essential extension

M ⊂ E with E injective is an injective envelope **of** M.

3.6.15. Proposition. Any module M has an injective envelope. If M ⊂ E, E ′ are

two injective envelopes, then there is an isomorphism f : E → E ′ fixing M.

Pro**of**. By 3.6.12 choose M ⊂ E ′ with E ′ injective. By Zorn’s lemma choose

M ⊂ E ⊂ E ′ maximal among the essential extensions **of** M. If E = E ′ then the

set **of** modules N ′ ⊂ E ′ such that E ∩ N ′ is nonempty by maximality **of** E. Let

N be maximal among these by Zorn’s lemma. It follows that E ⊕ N = E ′ and E

is injective by 3.6.6. Given two envelopes, let f : E → E ′ be any homomorphism

fixing M. Then f is injective, since M ⊂ E is essential. If f is not surjective, then

E ′ f(E) ⊕ E ′′ contradicting that M ⊂ E ′ is essential.

3.6.16. Exercise. (1) Let R be a domain. Show that the fraction field is an injective

module.

(2) Let R be a domain. Show that the torsion free divisible module injective.

(3) Show that for a ring that all modules are projective if and only all modules are injective.

α

3.7. Flat modules

3.7.1. Definition. An R-module F is a flat module if for any exact sequence 0 →

M → N the sequence

0 → M ⊗R F → N ⊗ F

is exact.

3.7.2. Example. If a ∈ R is a nonzero divisor and M is a flat modules, then aM

is injective and a is a nonzero divisor on M.

3.7.3. Proposition. A direct summand in a flat module is flat.

Pro**of**. Let F ⊕ F ′ be flat and M → N injective. Then M ⊗R (F ⊕ F ′ ) →

N ⊗R (F ⊕ F ′ ) is injective. Conclusion by 2.6.11.

3.7. FLAT MODULES 55

3.7.4. Proposition. Let Fα be family **of** flat modules, then the direct sum

α Fα

is a flat module.

Pro**of**. Let M → N be injective. Then M ⊗R ( Fα) → N ⊗R ( Fα) is

injective by 2.6.11 and 3.1.6, so the product is flat.

3.7.5. Example. A free module is flat.

3.7.6. Corollary. A projective module is flat.

Pro**of**. By 3.5.7 a projective module is a direct summand in a free. Conclusion by

3.7.4.

3.7.7. Proposition. Let F, F ′ be flat modules. Then F ⊗R F ′ is flat.

Pro**of**. Let M → N be injective. Then by 2.6.10 M ⊗R (F ⊗R F ′ ) → N ⊗R

(F ⊗R F ′ ) is (M ⊗R F ) ⊗R F ′ → (N ⊗R F ) ⊗R F ′ being injective by using the

definition twice.

3.7.8. Proposition. Let R → S be a ring homomorphism and F a flat R-module.

The change **of** ring module F ⊗R S is a flat S-module.

Pro**of**. Let M → N be an injective homomorphism **of** S-modules. Then by 2.7.5

M ⊗S (F ⊗R S ′ ) → N ⊗S (F ⊗R S) is M ⊗R F → N ⊗R F being injective since

F is a flat R-module.

3.7.9. Proposition. Let R be a ring and F a module. The following are equivalent.

(1) F is a flat module.

(2) HomR(F, E) is an injective module for any injective module E.

Pro**of**. Let M → N be injective. By 3.6.13 M ⊗R F → N ⊗R F is injective if and

only if HomR(N ⊗R F, E) → HomR(M ⊗R F, E) is surjective for any injective

module E. By 2.6.13 this is HomR(N, HomR(F, E)) → HomR(M, HomR(F, E)).

3.7.10. Corollary. Given a short exact sequence

0

M

N

F

where F is a flat module. Then M is flat if and only if N is flat.

Pro**of**. Let E be an injective module. By 3.7.9 and 3.6.3 the sequence

0

HomR(F, E)

HomR(N, E)

0

HomR(M, E)

is split exact. By 3.6.5 and 3.6.6 HomR(N, E) is injective if and only if HomR(M, E)

is so. Conclusion by 3.7.9.

3.7.11. Corollary. Given a short exact sequence

0

M

where F is a flat module. For any module L there is a short exact sequence

0

L ⊗R M

N

L ⊗R N

F

0

L ⊗R F

0

0

56 3. EXACT SEQUENCES OF MODULES

Pro**of**. Let E be an injective module. By 3.7.9 and 3.6.3 the sequence

0

HomR(F, E)

is split exact. So also the sequence

0

HomR(L, HomR(F, E))

HomR(N, E)

HomR(M, E)

HomR(L, HomR(N, E))

HomR(L, HomR(M, E))

is split exact. By 2.6.13 this is natural isomorphic to the sequence

0

Conclusion by 3.6.13.

HomR(L ⊗R F, E)

HomR(L ⊗R N, E)

HomR(L ⊗R M, E)

3.7.12. Proposition. A module F is flat, if for any ideal I ⊂ R

is exact.

0 → I ⊗R F → R ⊗R F

Pro**of**. By 3.7.9 it suffices to see that HomR(F, E) is injective for any injective E.

By 3.6.7 this amounts to HomR(R, HomR(F, E)) → HomR(I, HomR(F, E)) being

surjective. By 2.6.13 this homomorphism is HomR(R⊗RF, E) → HomR(I⊗R

F, E) which is surjective since E is injective.

3.7.13. Exercise. (1) Show that Z/(n) is not a flat Z-module for n = 0, 1.

(2) Show that Z/(2) is a flat Z/(6)-module.

(3) Show that Z/(2) is not a flat Z/(4)-module.

0

0

0

4

Fraction constructions

4.1. Rings **of** fractions

4.1.1. Lemma. Let R be a ring and U ⊂ R such that 1 ∈ U and for u, v ∈ U the

product uv ∈ U. On U × R is defined a relation

(u, a) ∼ (u ′ , a ′ ) ⇔ there is v ∈ U such that vu ′ a = vua ′

(1) The relation is an equivalence relation.

(2) If (u, a) ∼ (u ′ , a ′ ) and (v, b) ∼ (v ′ , b ′ ) then (uv, va + ub) ∼ (u ′ v ′ , v ′ a ′ +

u ′ b ′ ).

(3) If (u, a) ∼ (u ′ , a ′ ) and (v, b) ∼ (v ′ , b ′ ) then (uv, ab) ∼ (u ′ v ′ , a ′ b ′ ).

Pro**of**. The claims are proved by simple calculations. (1) Symmetry is clear. Reflexive

follows as 1 ∈ U. Transitive: if (u, a) ∼ (u ′ , a ′ ), (u ′ , a ′ ) ∼ (u ′′ , a ′′ ) then

vu ′ a = vua ′ , v ′ u ′′ a ′ = v ′ u ′ a ′′ . Since multiplication is commutative (v ′ vu ′ )u ′′ a =

v ′ u ′′ vua ′ = (v ′ vu ′ )ua ′′ give (u, a) ∼ (u ′′ , a ′′ ). (2) From wu ′ a = wua ′ , w ′ v ′ b =

w ′ vb ′ follow that w ′ vv ′ wu ′ a = w ′ vv ′ wua ′ , wuu ′ w ′ v ′ b = wuu ′ w ′ vb ′ . So now

ww ′ (u ′ v ′ )(va + ub) = ww ′ (uv)(v ′ a ′ + u ′ b ′ ) as needed. (3) Is similar.

4.1.2. Definition. Let R be a ring. U ⊂ R is a multiplicative subset if 1 ∈ U

and for u, v ∈ U the product uv ∈ U. The ring **of** fractions U −1R is given by

on U × R under the relation 4.1.1

equivalence classes a

u

(u, a) ∼ (u ′ , a ′ ) ⇔ there is v ∈ U such that vu ′ a = vua ′

The addition is

a b

+

u v

and the multiplication is

a b

·

u v

The canonical ring homomorphism is

= va + ub

uv

= ab

uv

ι : R → U −1 R, a ↦→ a

1

4.1.3. Proposition. Let φ : R → S be a ring homomorphism and U ⊂ R a

multiplicative subset. If all elements in φ(U) ⊂ S are units, then there exists a

unique ring homomorphism φ ′ : U −1 R → S such that φ = φ ′ ◦ ι.

φ

R

ι

U −1R Pro**of**. φ ′ ( a

u ) = φ(a)φ(u)−1 is the well defined unique ring homomorphism. Observe

that the elements **of** form bv −1 satisfy the rules for fractions.

57

φ ′

S

58 4. FRACTION CONSTRUCTIONS

4.1.4. Proposition. Let U ⊂ R be a multiplicative subset.

(1) a

1 = 0 in U −1R if and only if Ann(a) ∩ U = ∅.

(2) The canonical ring homomorphism R → U −1R is injective if and only if U

consists only **of** nonzero divisors.

Pro**of**. a

1 = 0 if and only if va = 0 for some v ∈ U.

4.1.5. Corollary. If R is a domain and U is the nonzero elements then K = U −1 R

is a field and the canonical homomorphism identifies R as a subring.

4.1.6. Definition. The field K in 4.1.5 is the fraction field **of** R.

4.1.7. Corollary. If R is a domain with fraction field K and φ : R → L is an

injective ring homomorphism into a field, then there is a unique homomorphism

K → L extending φ.

4.1.8. Definition. The total ring **of** fractions **of** R is U −1 R, where U is the set **of**

nonzero divisors **of** R.

4.1.9. Corollary. A ring is a subring **of** its total ring **of** fractions.

4.1.10. Proposition. Let U = {u n } the powers **of** an element u ∈ R. There is an

isomorphism

R[X]/(uX − 1) → U −1 R, X ↦→ 1

u

The ring **of** fractions with only one denominator is **of** finite type.

Pro**of**. A pair **of** inverse homomorphism are constructed by 1.6.7 and 4.1.3.

4.1.11. Exercise. (1) Show that if U contains a nilpotent element, then U −1 R = 0.

(2) Show that

Ker R → U −1 R = {a ∈ R|ua = 0, for some u ∈ U}

(3) Let U = {u1, . . . , um} and u = u1 · · · um. Then show that

U −1 R = {u n } −1 R

(4) Let R1, R2 be domains with fraction fields K1, K2. Show that the total ring **of** fractions

**of** R1 × R2 is K1 × K2.

(5) Show that the ring

{ a

∈ Q|a ∈ Z, n ∈ N}

2n is **of** finite type over Z.

4.2. Modules **of** fractions

4.2.1. Lemma. Let R be a ring and U ⊂ R multiplicative. On U × M is defined

a relation

(u, x) ∼ (u ′ , x ′ ) ⇔ there is v ∈ U such that vu ′ x = vux ′

(1) The relation is an equivalence relation.

(2) If (u, x) ∼ (u ′ , x ′ ) and (v, y) ∼ (v ′ , y ′ ) then (uv, vx + uy) ∼ (u ′ v ′ , v ′ x ′ +

u ′ y ′ ).

(3) If (u, a) ∼ (u ′ , a ′ ) in U × R 4.1.1 and (v, x) ∼ (v ′ , x ′ ) then (uv, ax) ∼

(u ′ v ′ , a ′ x ′ ).

Pro**of**. The claims are proved by simple calculations. See the pro**of** 4.1.1.

4.2. MODULES OF FRACTIONS 59

4.2.2. Definition. Let R be a ring and U a multiplicative subset. The module **of**

fractions U −1M is given by equivalence classes x

u on U × M under the relation

4.2.1

(u, x) ∼ (u ′ , x ′ ) ⇔ there is v ∈ U such that vu ′ x = vux ′

The addition is

x y

+

u v

and the U −1R-scalar multiplication is

a y

·

u v

The canonical homomorphism is

= vx + uy

uv

= ay

uv

M → U −1 M, x ↦→ x

1

4.2.3. Lemma. Let R be a ring and f : M → N a homomorphism **of** R-modules.

Then there is a homomorphism U −1f : U −1M → U −1N, x f(x)

u ↦→ u **of** U −1R modules.

Pro**of**. The claims are proved by simple calculations. If (u, x) ∼ (u ′ , x ′ ) then

vu ′ x = vux ′ and therefore vu ′ f(x) = vuf(x ′ ) showing (u, f(x)) ∼ (u ′ , f(x ′ )).

So the map is well defined. The rest is similar.

4.2.4. Proposition. The construction 4.2.2

and 4.2.3

M ↦→ U −1 M

f : M → N ↦→ U −1 f : U −1 M → U −1 N

is a functor from R-modules to U −1 R-modules.

Pro**of**. Follows from the definitions by simple calculations as in the pro**of** **of** 4.2.3.

For example, U −1 (f + g) = U −1f + U −1g, follows from f(x)+g(x)

u

g(x)

u .

= f(x)

u +

4.2.5. Remark. The induced homomorphism relates to the canonical homomorphism

such that the diagram is commutative.

M

f

N

U −1M U −1f

U −1N That is, the canonical homomorphism is a natural homomorphism.

4.2.6. Proposition. Let U ⊂ R be a multiplicative subset and M a module.

(1) x

1 = 0 in U −1M if and only if Ann(x) ∩ U = ∅.

(2) The canonical homomorphism M → U −1M is injective if and only if U

consists only **of** nonzero divisors on M.

Pro**of**. x

1 = 0 if and only if vx = 0 for some v ∈ U.

60 4. FRACTION CONSTRUCTIONS

4.2.7. Proposition. If Mα is a family **of** modules, then the homomorphism

U −1 (

Mα) →

is a natural isomorphism **of** U −1 R-modules.

α

α

U −1 Mα

Pro**of**. This is the method **of** common denominators in a finite sum.

xi

=

ui

1

(Πj=iuj)xi

Πiui

i

4.2.8. Exercise. (1) Show that if U contains a nilpotent element, then U −1 M = 0.

(2) Show that

Ker M → U −1 M = {x ∈ M|ux = 0, for some u ∈ U}

(3) Let U = {u1, . . . , um} and u = u1 · · · um. Then show that

U −1 M = {u n } −1 M

(4) Show that U −1 M = 0 if and only if U ∩ Ann(x) = ∅ for all x ∈ M.

(5) Show that the fraction homomorphism **of** a composition is the composition **of** the

respective fraction homomorphisms.

(6) Let M be a free R-module. Show that U −1 M is a free U −1 R-module

4.3. Exactness **of** fractions

4.3.1. Proposition. Let R be a ring and U a multiplicative subset. Given an exact

sequence **of** R-modules

M f

Then the following sequence is exact

U −1 M

i

g

N

U −1N

L

U −1L Pro**of**. If y

u ∈ U −1N maps to g(y)

u = 0 then there is v ∈ U such that 0 = vg(y) =

g(vy). Choose x ∈ M such that f(x) = vy. Then x

exactness.

4.3.2. Corollary. Given a short exact sequence

0

f

M

Then the following sequence is exact

0

U −1M g

N

U −1N

L

vu

maps to f(x)

vu

0

U −1L If the first sequence is split exact, also the second sequence is split exact.

0

= y

u proving

4.3.3. Corollary. For a homomorphism f : M → N there are natural isomorphisms

**of** U −1 R-modules.

(1) U −1 Ker f Ker U −1 f.

(2) U −1 Im f Im U −1 f.

(3) U −1 Cok f Cok U −1 f.

4.3. EXACTNESS OF FRACTIONS 61

Pro**of**. Represent the statements using short exact sequences. (1) The kernel is

determined by the exact sequence 0 → Ker f → M → N, 3.1.4. (3) The cokernel

is determined by the exact sequence M → N → Cok f → 0, 3.1.5. (2) The

image is determined by the exact sequence 0 → Ker f → M → Im f → 0, 2.3.5,

3.1.4.

4.3.4. Corollary. For submodules N, L ⊂ M there are natural identifications **of**

U −1 R-submodules and factor modules.

(1) U −1 (M/N) = U −1 M/U −1 N.

(2) U −1 (N + L) = U −1 N + U −1 L.

(3) U −1 (N ∩ L) = U −1 N ∩ U −1 L.

Pro**of**. Represent the statements using short exact sequences. (1) 0 → N → M →

M/N → 0 is short exact giving 0 → U −1 N → U −1 M → U −1 (M/N) → 0. The

wanted isomorphism follows form 3.1.5. (2) N + L is the image **of** N ⊕ L → M

so conclude by 4.3.3. (3) N ∩ L is the kernel **of** N ⊕ L → M so conclude by

4.3.3.

4.3.5. Corollary. For ideals I, J ⊂ R there are natural identifications in U −1 R.

(1) U −1 (R/I) = U −1 R/U −1 I.

(2) U −1 (I + J) = U −1 I + U −1 J.

(3) U −1 (I ∩ J) = U −1 I ∩ U −1 J.

(4) U −1 (IJ) = U −1 IU −1 J.

Pro**of**. (1) (2) (3) These are special cases **of** 4.3.4. (4) Both sides have the same

, a ∈ I, b ∈ J, u ∈ U.

generators ab

u

4.3.6. Proposition. Let R → U −1 R be the canonical homomorphism.

(1) For an ideal I ⊂ R the extended ideal

IU −1 R = U −1 I

(2) For an ideal J ⊂ U −1 R the extended contracted ideal

U −1 (J ∩ R) = J

(3) For an ideal I ⊂ R the contracted extended ideal

I ⊂ IU −1 R ∩ R

Pro**of**. (1) This is clear. (2) U −1 (J ∩ R) ⊂ J is true for any ring homomorphism

1.2.6. If b

b

1 b

u ∈ J then 1 ∈ J giving b ∈ J ∩ R and ub = u ∈ U −1 (J ∩ R). (3) This

is true for any ring homomorphism 1.2.6.

4.3.7. Proposition. Let R → U −1 R be the canonical homomorphism. For an

ideal I ⊂ R the contracted extended ideal

I = IU −1 R ∩ R

if and only if each u ∈ U is a nonzero divisor on R/I.

Pro**of**. Apply the snake lemma 3.2.4 to

0

0

I

0

R

U −1R/U −1I U −1R/U −1I

0

R/I

0

62 4. FRACTION CONSTRUCTIONS

and get the exact sequence

Conclusion from 4.2.6.

0 → I → IU −1 R ∩ R → Ker(R/I → U −1 (R/I)) → 0

4.3.8. Corollary. Let P ⊂ R be a prime ideal. Then U −1 P ⊂ U −1 R is either a

prime ideal or the whole ring.

Pro**of**. By 4.1.5 and 4.3.5 U −1 R/U −1 P = U −1 (R/P ) is either 0 or a domain.

4.3.9. Corollary. Let R be a principal ideal domain. Then U −1 R is a principal

ideal domain.

Pro**of**. A restricted ideal is principal by hypothesis and the extension **of** a principal

ideal is principal. Conclude by 4.3.6.

4.3.10. Proposition. Let R be a unique factorization domain. Then U −1 R is a

unique factorization domain.

Pro**of**. By 4.3.7 the extension **of** an irreducible element is either a unit or an irreducible

element. Since a principal ideal ( a a

u ) = ( 1 ), a factorization into irreducibles

in R gives a factorization in U −1R. Now if (a) = (p1) . . . (pn) is a factorization

in R and ( a

1 ) is irreducible in U −1R. Then all but one pi

1 is a unit in U −1R so

( a pi

1 ) = ( 1 ) is a prime ideal 4.3.9. The conditions 1.5.3 are satisfied.

4.3.11. Exercise. (1) Let U ⊂ R be multiplicative. Show that

U −1 (R[X]) = (U −1 R)[X]

(2) Let φ : R → S be a ring homomorphism and U ⊂ R a multiplicative subset. Show

that

U −1 S = φ(U) −1 S

4.4. Tensor modules **of** fractions

4.4.1. Proposition. Let R be a ring and U a multiplicative subset. For any module

M, the homomorphism

M ⊗R U −1 R → U −1 M, x ⊗ a ax

↦→

u u

is a natural isomorphism **of** U −1R-modules. Pro**of**. By 2.6.3 there is an R-module homomorphism x ⊗ a

u

b a

ba bax b ax

(x ⊗ ) = x ⊗ ↦→ =

v

u

vu

vu

map U −1 M → M ⊗R U −1 R, x

u

v

ax → u . By definition

u , so the this is a U −1 R-homomorphism. The

↦→ x ⊗ 1

u

is an inverse.

4.4.2. Remark. The two constructions, module change **of** ring to a fraction ring

and fraction module are natural isomorphic functors from modules to modules over

the fraction ring.

4.4.3. Corollary. Let R be a ring and U a multiplicative subset. Then U −1 R is a

flat R-module.

Pro**of**. This follows from 4.4.1 and 4.3.1.

4.4.4. Corollary. Let R be a ring and U a multiplicative subset and M, N modules.

Then there is a natural isomorphism

U −1 (M ⊗R N) U −1 M ⊗ U −1 R U −1 N

4.5. HOMOMORPHISM MODULES OF FRACTIONS 63

Pro**of**. This follows from 2.7.5 and 4.4.1.

4.4.5. Corollary. Let I ⊂ R be an ideal and U a multiplicative subset For a

module M module

U −1 (IM) U −1 IU −1 M

Pro**of**. Use that IM = Im(I ⊗R M → M) and 4.4.4.

4.4.6. Exercise. (1) Let M be a flat R-module. Show that U −1 M is a flat U −1 Rmodule.

(2) Let M be a projective R-module. Show that U −1 M is a projective U −1 R-module.

4.5. Homomorphism modules **of** fractions

4.5.1. Proposition. Let R be a ring and U a multiplicative subset. For any modules

M, N there is a natural homomorphism

**of** U −1 R-modules.

U −1 HomR(M, N) → Hom U −1 R(U −1 M, U −1 N)

Pro**of**. Given f : M → N and u ∈ U the setting x

homomorphism U −1 M → U −1 N.

v ↦→ f(x)

uv is a U −1R 4.5.2. Proposition. Let R be a ring and U a multiplicative subset. For any Rmodule

M and any U −1 R-modules N there is a natural isomorphism

**of** U −1 R-modules.

HomR(M, N) → Hom U −1 R(U −1 M, N)

Pro**of**. This is the change **of** rings isomorphism 2.7.6 interpreted according to

4.4.2.

4.5.3. Example. Let R be a ring and U a multiplicative subset. R → U −1 R the

canonical homomorphism. The induced module functor maps an R-module M to

the U −1 R-module HomR(U −1 R, M). The natural isomorphism 2.7.10 is

for any U −1 R-module N.

HomR(N, M) Hom U −1 R(N, HomR(U −1 R, M))

4.5.4. Example. The homomorphism 4.5.1 is in general neither injective nor surjective.

(1) Not surjective:

(2) Not injective:

0 = HomZ(Q, Z) → HomQ(Q, Q) = Q

0 = HomZ(Q, Q/Z) ⊗Z Q → HomQ(Q, Q/Z ⊗Z Q) = 0

4.5.5. Exercise. (1) Let M be a Z-module and N a Q-module. Show that there is an

isomorphism HomZ(M, N) HomQ(M ⊗Z Q, N).

64 4. FRACTION CONSTRUCTIONS

4.6. The polynomial ring is factorial

4.6.1. Definition. Let R be a unique factorization domain and let f = anX n +

· · · + a0 be a polynomial over R. Then the content **of** polynomial f, c(f), is the

greatest common divisor **of** the coefficients a0, . . . , an.

4.6.2. Proposition (Gauss’ lemma). Let R be a unique factorization domain. For

polynomials f, g ∈ R[X]

c(fg) = c(f)c(g)

Pro**of**. Assume by cancellation that c(f), c(g) are units in R. For any irreducible

p ∈ R the projections **of** f, g in R/(p)[X] are nonzero. Since R has unique factorization

the ideal (p) is a prime ideal. It follows that the projection **of** the product

fg in R/(p)[X] is also nonzero and therefore p is not a common divisor **of** the

coefficients **of** the product fg.

4.6.3. Proposition. Let R be a unique factorization domain. Then the ring **of**

polynomials R[X] is a unique factorization domain.

Pro**of**. Let K be the fraction field **of** R, then the polynomial ring K[X] is a principal

ideal domain. Let f ∈ R[X] and use unique factorization in K[X] to get

0 = a ∈ R and p1, . . . , pn ∈ R[X], irreducible in K[X], such that

af = p1 . . . pn

Assume by 4.6.2 that a = 1 and c(p1), . . . , c(pn) are units in R. Apply 4.6.2 and

1.6.5 to see that p1, . . . , pn are irreducible in R[X]. An irreducible p ∈ R generates

a prime ideal (p) ⊂ R[X]. A non constant irreducible p ∈ R[X] generates a prime

ideal (p) ⊂ K[X] and therefore also a prime ideal (p) ⊂ R[X]. So conditions

1.5.3 are satisfied.

4.6.4. Theorem. Let K be a field. Then the polynomial ring K[X1, . . . , Xn] is a

unique factorization domain.

Pro**of**. Follows by induction from 4.6.3.

4.6.5. Exercise. (1) Let f ∈ Z[X] be monic and assume f = gh where g, h ∈ Q[X]

are monic. Show that g, h ∈ Z[X].

(2) Let f ∈ Z[X] be monic and irreducible in Z/(n)[X]. Show that f is irreducible

Q[X].

(3) Let K be a field. Show that the polynomial ring K[X1, X2, . . . ] in countable many

variables is a unique factorization domain

5

Localization

5.1. Prime ideals

5.1.1. Theorem (Krull). A nonzero ring contains a maximal ideal.

Pro**of**. The nonempty set **of** ideals different from R is ordered by inclusion. Given

an increasing chain Iα then ∪Iα is an ideal different from R which is a maximum

for the chain. Conclusion by Zorn’s lemma.

5.1.2. Corollary. Any proper ideal in a ring is contained in a maximal ideal.

Pro**of**. The factor ring is nonzero and contains a maximal ideal. Conclusion by

1.2.10.

5.1.3. Proposition. Let P1, . . . , Pn ⊂ R be ideals with at most 2 not being prime

ideals. If an ideal

I ⊂ P1 ∪ · · · ∪ Pn

then I ⊆ Pi for some i.

Pro**of**. Assume n > 1 and I is not contained in any sub union. Moreover assume

the numbering such that P3, . . . , Pn are prime ideals. Then for each i there is

The element

ai ∈ (I ∩ Pi)\ ∪j=i Pj

an + a1 . . . an−1

is in I but not in any Pi, giving a contradiction. So n = 1.

5.1.4. Proposition. Let P1, . . . , Pn ⊂ R be prime ideals and I any ideal. If for

some a ∈ R

a + I ⊂ P1 ∪ · · · ∪ Pn

then I ⊆ Pi for some i.

Pro**of**. If a ∈ ∩iPi then conclusion by 5.1.3. On the contrary after renumbering

there exists j with 1 ≤ j < n such that

a ∈ P1 ∩ · · · ∩ Pj\Pj+1 ∪ · · · ∪ Pn

Assume no inclusions between the prime ideals and choose by 5.1.3

b ∈ I ∩ Pj+1 ∩ · · · ∩ Pn\P1 ∪ · · · ∪ Pj

Then a + b /∈ ∪iPi contradicts the hypothesis.

5.1.5. Proposition. Let R → U −1 R be the canonical homomorphism. Extension

and contraction gives a bijective correspondence between prime ideals in R disjoint

from U and all prime ideals in U −1 R.

(1) For a prime ideal P ⊂ R\U the extended ideal P U −1 R is a prime ideal in

U −1 R and the contracted P U −1 R ∩ R = P .

65

66 5. LOCALIZATION

(2) For a prime ideal Q ⊂ U −1 R the contracted ideal Q ∩ R is a prime ideal

and the extended (Q ∩ R)U −1 R = Q

Pro**of**. Conclusion by 4.3.6, 4.3.7, 4.3.8.

5.1.6. Corollary. Let R be a ring and U a multiplicative subset.

(1) An ideal maximal among the ideals disjoint from U is a prime ideal.

(2) Any ideal disjoint from U is contained in a prime ideal disjoint form U.

Pro**of**. The prime ideals disjoint from U are the prime ideals in U −1 R.

5.1.7. Proposition. The nilradical **of** a ring R is the intersection **of** all prime ideals

P .

√

0 = P

Pro**of**. By 1.3.8 the nilradical is contained in any prime ideal. Suppose u ∈ R is

not nilpotent. Then {u n } −1 R is nonzero. Then contraction **of** a maximal ideal,

5.1.1, is a prime ideal in R not containing u.

5.1.8. Corollary. Let R be a ring.

(1) The radical **of** an ideal I is the intersection **of** all prime idealsP containing I

√

I = P

P

I⊂P

(2) For ideals I, J ⊂ R, √ I ∩ J = √ I ∩ √ J.

(3) If U is a multiplicative subset, then U −1√ 0 = √ 0 in U −1 R.

If R is reduced, then U −1 R is reduced.

Pro**of**. (1) Use 5.1.7 on the factor ring R/I. (2) This follows from (1). (3) Use the

correspondence 5.1.5.

5.1.9. Definition. A prime ideal minimal for inclusion among prime ideals is a

minimal prime ideal.

5.1.10. Proposition. Any prime ideal **of** Q ⊂ R contains a minimal prime ideal

P ⊂ Q.

Pro**of**. The set **of** prime ideals in R is ordered by inclusion. Given a decreasing

chain Pα then ∩Pα is a prime ideal. Conclusion by Zorn’s lemma.

5.1.11. Corollary. Let R ⊂ S be a subring and P ⊂ R a minimal prime ideal.

Then there is a minimal prime ideal Q ⊂ S contracting to P = Q ∩ R.

5.1.12. Exercise. (1) Let K ⊂ R be an infinite subfield and I, P1, . . . , Pn any ideals.

Show that if I ⊂ P1 ∪ · · · ∪ Pn then I ⊂ Pi for some i.

(2) Let P, P1, P2 be proper ideals. Show that if P is a maximal ideal and P n ⊂ P1 ∪ P2

then P = P1 **of** P = P2.

5.2. LOCALIZATION OF RINGS 67

5.2. Localization **of** rings

5.2.1. Definition. A ring R which contains precisely one maximal ideal P is a

local ring and denoted (R, P ). The residue field **of** R is R/P denoted by k(P ).

A ring homomorphism φ : R → S **of** local rings (R, P ), (S, Q) is a local ring

homomorphism if φ(P ) ⊂ Q.

5.2.2. Lemma. Let R be a ring and Q = ∅ a subset. Then R is a local ring with

maximal ideal Q if and only if R\Q is the set **of** units in R.

Pro**of**. Use that an ideal I = R if and only if it contains a unit.

5.2.3. Lemma. A ring homomorphism φ : R → S **of** local rings (R, P ), (S, Q) is

a local ring homomorphism if the extended ideal P S ⊂ Q or the contracted ideal

Q ∩ R = P . The residue homomorphism k(P ) → k(Q) is a field extension.

Pro**of**. The contraction Q ∩ R is a prime ideal containing P . The rest is clear.

5.2.4. Lemma. Let R be a ring and P a prime ideal. Then U = R\P is a multiplicative

subset. The ring **of** fractions U −1 R is a local ring. The maximal ideal is

the extended ideal P U −1 R. The residue field is U −1 R/P U −1 R which is canonical

isomorphic to the fraction field **of** R/P .

5.2.5. Definition. Let R be a ring, P a prime ideal and U = R\P . The localized

ring at P is the local ring RP = U −1 R. The residue field is denoted k(P ) =

RP /P RP .

5.2.6. Example. Let the ring be Z.

(1) The local ring at (0) is the fraction field Q = Z (0).

(2) The local ring Z (p) for a prime number p is identified with a subring **of** Q

Z (p) = { m

|p not dividing n}

n

The residue field Fp = Z (p)/(p).

(3) Any nonzero ideal in Z (p) is principal **of** the form (pn ) for some n.

5.2.7. Proposition. Let (R, P ) be a local ring. One **of** the following conditions is

satisfied:

(1) The characteristic char(R) = 0. P ∩ Z = (0) and Q ⊂ R is a subfield.

Q → k(P ) is a field extension.

(2) The characteristic char(R) = 0. P ∩ Z = (p), p a prime number. Z (p) ⊂ R

is a local subring. Fp → k(P ) is a field extension.

(3) The characteristic char(R) = p n , a power **of** a prime number. Z/(p n ) ⊂ R

is a local subring. Fp → k(P ) is a field extension.

Pro**of**. (1) (2) are clear by 5.2.3 and 5.2.6. (3) If the characteristic is nonzero then

any prime ideal contracts Q ∩ Z = (p). So a prime number q = p gives a unit in

R. There is a local homomorphism Z (p) → R, 4.1.3. The nontrivial kernel is (p n )

by 5.2.6.

5.2.8. Example. A field K is a local ring with maximal ideal (0). The power series

ring K[[X]] is a local ring with maximal ideal (X) and residue field K.

5.2.9. Remark. Let R be a domain.

(1) The local ring at (0) is the fraction field K = R (0).

68 5. LOCALIZATION

(2) Any local ring RP is identified with a subring **of** the fraction field K.

(3) The intersection

R =

RP , P a maximal ideal

P

5.2.10. Remark. Let R × S be a product **of** rings.

(1) A prime ideal is **of** the form P × S or R × Q for uniquely determined prime

ideals P ⊂ R or Q ⊂ S.

(2) The local ring at P ×S is identified with RP through the projection R ×S →

R.

(3) The local ring at R × Q is identified with SQ through the projection R × S →

S.

5.2.11. Proposition. Let P be a prime ideal and R → RP the canonical homomorphism.

Extension and contraction gives a bijective correspondence between

prime ideals in R contained in P and all prime ideals in RP .

(1) For a prime ideal Q ⊂ P the extended ideal QRP is a prime ideal in RP and

the contracted QRP ∩ R = Q.

(2) For a prime ideal Q ′ ⊂ RP the contracted ideal Q ′ ∩ R ⊂ P is a prime ideal

and the extended (Q ′ ∩ R)RP = Q ′

Pro**of**. This is a special case **of** 5.1.5.

5.2.12. Lemma. Let Q ⊂ P ⊂ R be prime ideals. Then QRP is a prime ideal in

RP and canonically

Pro**of**. By fraction rules a w av

u / v = uw .

RQ = (RP )QRP

5.2.13. Definition. The intersection **of** all maximal ideals in a ring is the Jacobson

radical.

5.2.14. Remark. The Jacobson radical contains the nilradical. In a local ring the

Jacobson radical is the maximal ideal.

5.2.15. Exercise. (1) Show that a local ring is never a product **of** two nonzero rings.

(2) Show that a ∈ R is in the Jacobson radical if and only if 1+ab is a unit for all b ∈ R.

(3) Let p be a prime number. Describe the prime ideals in the ring Z (p).

(4) Let P be a prime ideal. Show that k(P ) is the fraction field **of** R/P .

(5) Let (R, P ) be a local ring. Show that (R[[X]], (P, X) is a local ring and the canonical

homomorphism R → R[[X]] is a local homomorphism.

5.3. Localization **of** modules

5.3.1. Definition. Let R be a ring, P a prime ideal and U = R\P . For a module

M, the localized module at P is the module MP = U −1 M over the local ring RP .

For a homomorphism f : M → N the localized homomorphism is fP : MP →

NP and the residue homomorphism is f(P ) : M ⊗R k(P ) → N ⊗R k(P ).

The constructions are functors, 2.7.4, 4.2.4.

5.3.2. Lemma. Let Q ⊂ P ⊂ R be prime ideals and M an R-module Then QRP

is a prime ideal in RP and canonically

Pro**of**. See pro**of** **of** 5.2.12.

MQ = (MP )QRP

5.3. LOCALIZATION OF MODULES 69

5.3.3. Proposition. Let R be a ring and P a prime ideal. If Mα is a family **of**

modules, then the homomorphism

(

Mα)P →

(Mα)P

is an isomorphism **of** RP -modules.

Pro**of**. See 4.2.7.

5.3.4. Corollary. For a homomorphism f : M → N

(1) (Ker f)P Ker fP .

(2) (Im f)P Im fP .

(3) (Cok f)P Cok fP .

Pro**of**. See 4.3.3.

α

5.3.5. Corollary. Let R be a ring and P a prime ideal. For submodules N, L ⊂ M

(1) (M/N)P MP /NP .

(2) (N + L)P NP + LP .

(3) (N ∩ L)P NP ∩ LP .

Pro**of**. See 4.3.4.

5.3.6. Proposition. Let R be a ring, P a prime ideal and M a module.

(1) MP M ⊗R RP .

(2) MP /P RP MP M ⊗R k(P ).

Pro**of**. See 4.4.1.

5.3.7. Proposition. Let R be a ring, P a prime ideal.

(1) For an R module M and an RP -module N there is a natural isomorphism

M ⊗R RP ⊗RP N M ⊗R N

(2) For an R module M, L there is a natural isomorphism

Pro**of**. See 4.2.7 and 2.7.4.

(M ⊗R L)P MP ⊗RP LP

5.3.8. Definition. Let R be a ring. F is a locally free module is FP is a free

RP -module for all prime ideals P .

5.3.9. Lemma. Let R be a ring. F is a locally free module if FQ is a free RQmodule

for all maximal ideals Q.

Pro**of**. A prime ideal P ⊂ Q is contained in a maximal ideal. By 5.3.6 FP

(FQ)PQ is free.

5.3.10. Example. A free module is a locally free module.

5.3.11. Exercise. (1) Let P ⊂ R be a prime ideal and R → S a ring homomorphism.

Show that RP → SP is a ring homomorphism.

(2) Let Q ⊂ S be a prime ideal and R → S a ring homomorphism. Show that RQ∩R →

SQ is a local ring homomorphism.

(3) Let R = K × L be a product **of** fields. Show that ideal K × {0} is locally free but

not free.

α

70 5. LOCALIZATION

5.4. Exactness and localization

5.4.1. Proposition. Let R be a ring and M a module. The following conditions

are equivalent.

(1) M = 0.

(2) MP = 0 for all prime ideals P .

(3) MP = 0 for all maximal ideals P .

Pro**of**. (1) ⇒ (2) ⇒ (3) is clear. (3) ⇒ (1): Let 0 = x ∈ M be given. Then

Ann(x) ⊂ P is contained in a maximal ideal, 5.1.2. Clearly 0 = x

1 ∈ MP

contradicts (3).

5.4.2. Corollary. Let R be a ring and f : M → N a homomorphism. The following

conditions are equivalent.

(1) f is injective.

(2) fP is injective for all prime ideals P .

(3) fP is injective for all maximal ideals P .

Pro**of**. Use 5.4.1 on Ker f.

5.4.3. Corollary. Let R be a ring and f : M → N a homomorphism. The following

conditions are equivalent.

(1) f is surjective.

(2) fP is surjective for all prime ideals P .

(3) fP is surjective for all maximal ideals P .

Pro**of**. Use 5.4.1 on Cok f.

5.4.4. Corollary. Let R be a ring and f : M → N a homomorphism. The following

conditions are equivalent.

(1) f is an isomorphism.

(2) fP is an isomorphism for all prime ideals P .

(3) fP is an isomorphism for all maximal ideals P .

5.4.5. Corollary. Let R be a ring and

0

f

M

g

N

a sequence **of** homomorphisms. The following conditions are equivalent.

(1) The sequence is short exact.

(2) The sequence

0

MP

fP

NP

is short exact for all prime ideals P .

(3) The sequence

0

MP

fP

NP

is short exact for all maximal ideals P .

L

gP

LP

gP

LP

5.4.6. Corollary. Let R be a ring and F a module. The following conditions are

equivalent.

(1) F is flat.

(2) FP is flat for all prime ideals P .

0

0

0

(3) FP is flat for all maximal ideals P .

5.5. FLAT RING HOMOMORPHISMS 71

Pro**of**. Let 0 → M → N. Use 5.3.7 and 5.4.2 on M ⊗R F → N ⊗R F .

5.4.7. Proposition. Let R be a ring and M a module. Then there is an exact

sequence

0 → M →

P maximal

Pro**of**. Let 0 = x ∈ M be given. Then Ann(x) ⊂ P is contained in a maximal

ideal, 5.1.2. Clearly 0 = x

1 ∈ MP .

5.4.8. Corollary. Let R be a ring. Then there is an injective ring homomorphism

R →

P maximal

5.4.9. Corollary. Let R be a ring. The following conditions are equivalent.

(1) R is reduced.

(2) RP is reduced for all prime ideals P .

(3) RP is reduced for all maximal ideals P .

Pro**of**. Use 5.1.8 and 5.4.10.

5.4.10. Exercise. (1) Let R be a ring and

0

f

M

g

N

a split exact sequence. Show that the localized sequence is split exact for all prime

ideals P .

RP

MP

L

5.5. Flat ring homomorphisms

5.5.1. Definition. A flat R-module F is faithfully flat if for any M = 0 the tensor

product M ⊗R F = 0.

5.5.2. Example. A nonzero free module is faithfully flat.

5.5.3. Lemma. Let F be a faithfully flat R-module and M → N a homomorphism.

If M ⊗R F → N ⊗R F is injective, then M → N is injective.

Pro**of**. Let f : M → N. Then 0 → Ker f ⊗R F → M ⊗R F → N ⊗R F is exact.

It follows, that Ker f = 0.

5.5.4. Definition. A ring homomorphism R → S is a flat ring homomorphism if

S is a flat R-module and a faithfully flat ring homomorphism if S is faithfully flat.

5.5.5. Proposition. Let R → S be a faithfully flat ring homomorphism. for any

ideal I ⊂ R the extended contracted returns I, i.e.

I = IS ∩ R

Pro**of**. Tensor the homomorphism R/I → R/IS∩R with S. The induced R/I⊗R

S → R/IS ∩ R ⊗R S is canonically isomorphic to the identity S/IS → S/IS.

By 5.5.3 R/I → R/IS ∩ R is injective giving I = IS ∩ R.

5.5.6. Corollary. A faithfully flat ring homomorphism R → S is injective.

5.5.7. Proposition. Let φ : R → S be a ring homomorphism. The following

conditions are equivalent

0

72 5. LOCALIZATION

(1) φ is flat.

(2) φP : RP → SP is flat for all prime ideals P ⊂ R.

(3) φP : RP → SP is flat for all maximal ideals P ⊂ R.

Pro**of**. Use 5.4.6.

5.5.8. Proposition. A flat local homomorphism (R, P ) → (S, Q) is faithfully flat.

Pro**of**. Let 0 = x ∈ M be an R-module. R/ Ann(x) Rx and I = Ann(x) ⊂ P ,

so IS ⊂ Q. Then R/I ⊗R S S/IS = 0 and therefore M ⊗R S = 0 giving the

claim.

5.5.9. Proposition (going-down). Let φ : R → S be a flat ring homomorphism

and Q ⊂ S a prime ideal. For any prime ideal P ′ ⊂ P = Q ∩ R there is a prime

ideal Q ′ ⊂ Q contracting to P ′ = Q ′ ∩ R.

Pro**of**. The local homomorphism RP → SQ is faithfully flat. The ring k(P ′ ) ⊗R

SQ is nonzero and therefor contains a maximal ideal Q ′′ . The contraction to S,

Q ′ = Q ′′ ∩ S contracts to P ′ = R ∩ Q ′ .

5.5.10. Proposition. Let R → S be a flat homomorphism. The following conditions

are equivalent.

(1) R → S is faithfully flat.

(2) Any prime ideal P ⊂ R is the contraction P = Q ∩ R **of** a prime ideal

Q ⊂ S.

Pro**of**. Assume (2). Let M = 0 be an R-module. Then MP = 0 for some P . Let

P = Q ∩ R, then (M ⊗R S)Q MP ⊗RP SQ = 0 as RP → SQ is faithfully flat.

So (1) is true.

5.5.11. Proposition. The inclusion R → R[X1, . . . , Xn] is a faithfully flat homomorphism

Pro**of**. The R-module R[X1, . . . , Xn] is free.

5.5.12. Exercise. (1) Show that a free module is faithfully flat.

(2) Let R → S and S → T be flat homomorphisms. Show that the composite R → S is

flat.

(3) Show that Q is a flat but not faithfully flat Z-module.

(4) Let R be a ring and I = √ 0 the nilradical. Show that IR[X] is the nilradical **of**

R[X].

6

Finite modules

6.1. Finite Modules

6.1.1. Definition. Let R be a ring. A finite module is generated by finitely many

elements. The finite free module with standard basis e1, . . . , en is denoted R n .

6.1.2. Lemma. Let R be a ring and M a module. The following are equivalent.

(1) M is generated by n elements x1, . . . , xn.

(2) There is a surjective homomorphism R n → M → 0, ei ↦→ xi.

Pro**of**. See 2.4.12.

6.1.3. Proposition. Let R → S be a ring homomorphism. If an R-module M is

generated by n elements x1, . . . , xn. Then the change **of** rings S-module M ⊗R S

is generated by x1 ⊗ 1, . . . , xn ⊗ 1 over S.

Pro**of**. Follows from 6.1.2 and 3.4.1

6.1.4. Corollary. Let R be a ring and U a multiplicative subset. If M is a finite

R-module, then U −1 M is a finite U −1 R-module.

6.1.5. Proposition. For a short exact sequence

0

f

M

the following h**old**

(1) If N is finite, then L is finite.

(2) If M, L are finite, then N is finite.

g

N

Pro**of**. (1) If y1, . . . , yn generates N, then g(y1), . . . , g(yn) generates L. (2) Choose

u : R n → M → 0 and v : R m → L → 0 exact. By 3.5.4 there is w : R m → N

such that g ◦ w = v. There is a diagram

0

0

Rn

u

M

Conclusion by the snake lemma 3.2.4.

6.1.6. Corollary. Let

0

f

f

M

L

Rn ⊕ Rm

f◦u+w

g

N

g

N

0

Rm

v

L

be a split exact sequence. Then N is finite if and only if M, L are finite.

Pro**of**. Let u be a retraction **of** f. By 6.1.5 Im u = M is finite. The rest is contained

in 6.1.5.

73

L

0

0

0

74 6. FINITE MODULES

6.1.7. Corollary. Let f : M → N be a homomorphism.

(1) If M is finite, then Im f is finite.

(2) If Ker f, Im f are finite, then M is finite.

(3) If N is finite, then Cok f is finite.

(4) If Im f, Cok f are finite, then N is finite.

Pro**of**. Use 6.1.5 on the exact sequences 3.1.8.

6.1.8. Corollary. Let M, N be modules. Then M ⊕ N is finite if and only if M

and N are finite.

Pro**of**. Use 6.1.6.

6.1.9. Proposition. Let R be a ring and M, N finite modules. Then M ⊗R N is

finite.

Pro**of**. Use 6.1.2 and 6.1.5. Let R m → M and R n → N be surjective. Then

R m ⊗R R n → M ⊗R N is surjective, 3.4.1.

6.1.10. Proposition. Let R be a ring and M a module. The following are equivalent.

(1) M is finite and projective.

(2) M is a direct summand in a finite free module.

Pro**of**. Use 6.1.2 and 6.1.8.

6.1.11. Proposition. Let R be a ring and M, N finite modules.

(1) If M is projective, then HomR(M, N) is finite.

(2) If M, N are projective, then HomR(M, N) is projective.

Pro**of**. Use 6.1.8.

6.1.12. Proposition. Let F be a finite projective module and E an injective module.

(1) F ⊗R E is injective.

(2) HomR(F, E) is injective.

Pro**of**. (1) (2) Both modules become summands in injective modules.

6.1.13. Proposition. Let R be a ring and U a multiplicative subset. For a finite

module M the following h**old**:

(1) U −1 M = 0 if and only if there is a u ∈ U such that uM = 0.

(2) Ann(U −1 M) = U −1 Ann(M) in U −1 R.

Pro**of**. Let x1, . . . , xn generate M. (1) U −1 M = 0 if and only if u1x1 = · · · =

unxn. Put u = u1 . . . un. (2) Ann(M) = Ann(x1) ∩ · · · ∩ Ann(xn). Now use (1)

and 4.3.4.

6.1.14. Proposition. Let R be a ring and Mα a family **of** modules. For any finite

module N there is a natural isomorphism

HomR(N,

Mα)

HomR(N, Mα)

α

Pro**of**. A homomorphism f : N → Mα has an image in a finite sum.

6.1.15. Exercise. (1) Show that

n Z/(n) is not a finite Z-module.

α

6.2. FREE MODULES 75

(2) Let K be a field and R = K[X1, X2, . . . ] the polynomial ring in countable many

variables. Show that R is a finite module, but the ideal (X1, X2, . . . ) is not a finite

module.

6.2. Free Modules

6.2.1. Definition. Let R be a ring and let R n be the free module with standard

basis e1, . . . , en.

(1) Let A = (aij) be a m × n-matrix with m rows and n columns, where the

entry aij ∈ R. Identify Rn with n-columns. Then matrix multiplication

x = (xj) ↦→ y = Ax, yi =

j

aijxj

defines a homomorphism R n → R m .

(2) Let f : R n → R m be a homomorphism. Then define a m × n-matrix A =

(aij) by

f(ej) =

i

aijei

6.2.2. Proposition. (1) The dictionary defined in 6.2.1 gives a canonical isomorphism

between the module **of** m × n-matrices and HomR(R n , R m ).

(2) Matrix multiplication corresponds to composition **of** homomorphisms and the

identity matrix corresponds to the identity homomorphism.

(3) Invertible matrices correspond to isomorphisms.

Pro**of**. Do linear **algebra** homework.

6.2.3. Definition. Let R be a ring.

(1) Let A = (aij) be a m × n-matrix. The (m − 1) × (n − 1) matrix derived

from A be deleting i-row and j-column is Aij.

(2) For a square matrix A the determinant is defined by row expansion and induction:

det(a11) = a11

det A =

i

(−1) 1+j a1j det A1j

(3) The determinant **of** a k × k-matrix derived from A by choosing entries from

k rows and columns is a k-minor **of** A.

(4) If A is a n × n-matrix, then the c**of**actor matrix A ′ = (a ′ ij ) has entries

given by (n − 1)-minors.

a ′ ij = (−1) i+j det Aji

6.2.4. Proposition. (1) The determinant **of** the identity matrix is 1.

(2) The determinant is calculated by any expansion

det A =

(−1) i+j aij det Aij =

i

(3) If A, B are n × n-matrices then the product rule h**old**s

det AB = det A det B

j

(−1) i+j aij det Aij

76 6. FINITE MODULES

(4) Let A be an n × n-matrix with c**of**actor matrix A ′ . Then the matrix Cramer’s

rule h**old**s

AA ′ = A ′ A = det A(1n)

Where (1n) is the n × n identity matrix.

(5) A square matrix A is invertible if and only if det A is a unit in R.

Pro**of**. More linear **algebra** homework.

6.2.5. Proposition. Let f : R n → R n be a homomorphism represented by an

n × n-matrix A.

(1) f is a surjective if and only if det A is a unit.

(2) f is injective if and only if det A is a nonzero divisor.

Pro**of**. (1) If f is surjective, then a section is represented by a matrix B such that

AB = (1n). Then det A is a unit. Conversely by 6.2.4. (2) If det A is a nonzero

divisor, then by 6.2.4 A ′ A = det A(1n) gives an injective homomorphism, so f is

injective. If det A is a zero divisor, then there is number k < n such that Ann(1 −

minors) = · · · = Ann(k − minors) = 0 and 0 = b ∈ Ann((k + 1) − minors).

Assume that the k−minor from first k rows and columns ck+1 has ck+1b = 0. Let

cj(−1) k+1+j be the k-minor from first k rows and first k + 1 columns excluding

number j and put cj = 0, j > k + 1. Then A(cj) is a column with entries being

(k + 1) − minors so f((bcj)) = A(bcj) = 0 and f is not injective.

6.2.6. Proposition. Let R be a ring and f : R n → R m homomorphism.

(1) If f is surjective, then n ≥ m.

(2) If f is injective, then n ≤ m.

Pro**of**. (1) If n < m let p : R m → R n be the projection onto first n coordinates.

Then f ◦p is surjective and represented by an m×m-matrix A with m-column zero.

A section to f ◦ p is represented by an m × m-matrix B such that BA = (1m). But

the product AB must have a zero m-column, so the contradiction gives n ≥ m.

(2) If n > m let i : R m → R n be injection onto first m coordinates. Then i ◦ f is

injective and represented by an n × n-matrix A with n-row zero. Then det A = 0

contradicting 6.2.5. So n ≤ m.

6.2.7. Proposition. (1) A finite free module has a finite basis.

(2) The number **of** elements in a basis for a finite free module is independent **of**

the basis.

Pro**of**. Let F be finite free generated by n elements. If y1, . . . , ym is part **of** a

basis, then by projection F → R m there is a surjective homomorphism R n → R m .

Conclusion by 6.2.6.

6.2.8. Definition. The number **of** elements 6.2.7 in a basis for a finite free module

F is the rank, rankR F .

6.2.9. Proposition. If x1, . . . , xn generates a free module F **of** rank n, then they

constitutes a basis.

Pro**of**. Choose a basis and an isomorphism f : R n → F . The homomorphism

g : R n → F, ei ↦→ xi is surjective. The composite f −1 ◦ g : R n → R n is

surjective and therefore by 6.2.5 an isomorphism. Then g is an isomorphism and

xi a basis.

6.3. CAYLEY-HAMILTON’S THEOREM 77

6.2.10. Proposition. Let F, F ′ be finite free modules. Then

(1) F ⊕ F ′ is free and rankR F ⊕ F ′ = rankR F + rankR F ′ .

(2) F ⊗R F ′ is free and rankR F ⊗R F ′ = rankR F · rankR F ′ .

(3) HomR(F, F ′ ) is free and rankR HomR(F, F ′ ) = rankR F · rankR F ′ .

6.2.11. Exercise. (1) Let R n → R n be a surjective homomorphism. Show that it is an

isomorphism.

6.3. Cayley-Hamilton’s theorem

6.3.1. Remark. Let R be a ring and f : M → M a homomorphism. By 2.1.13

view M as an R[X]-module, where Xx = f(x) for x ∈ M. The homomorphism

1.6.7, 2.6.9, R[X] → HomR(M, M), a ↦→ aM, X ↦→ f is a ring homomorphism.

The image is R[f] the smallest subring containing 1M, f. M is naturally a R[f]module

and the R[X]-module above is the restriction **of** scalars.

6.3.2. Proposition. Let A be an n × n-matrix and I the ideal generated by the

entries aij. The polynomial

det(X1n − A) = a0 + a1X + . . . an−1X n−1 + X n

has a0, . . . , an−1 ∈ I and gives the relation

as n × n-matrix.

a0(1n) + a1A + . . . an−1A n−1 + A n = 0

Pro**of**. View R n as a module over the ring R[X], 6.3.1, with scalar multiplication

Xx = Ax, x ∈ R n

Let the n × n-matrix U = X(1n) − A over R[X] have c**of**actor matrix U ′ . The

relations above give U ′ Uej = 0, written out by 6.2.4

det U ej = 0

for all j. That is det U M = 0. By calculation

in R[X], with ai ∈ I.

det U = a0 + a1X + · · · + an−1X n−1 + X n

6.3.3. Proposition. Let I ⊂ R be an ideal and f : M → M a homomorphism

on a finite module generated by n elements. Suppose Im f ⊂ IM, then there exist

a0, . . . , an−1 ∈ I such that

in HomR(M, M).

a01M + a1f + . . . an−1f n−1 + f n = 0

Pro**of**. Let x1, . . . , xn generate M and write

f(xj) =

i

aijxi

for an n × n-matrix A with entries aij ∈ I. View this over the ring R[X], 6.3.1.

Then

Xxj =

i

aijxi

78 6. FINITE MODULES

Let the n × n-matrix U = X(1n) − A over R[X] have c**of**actor matrix U ′ . The

relations above give U ′ Uxj = 0, written out by 6.2.4

det U xj = 0

for all j. That is det U M = 0. By calculation

in R[X], with ai ∈ I.

det U = a0 + a1X + · · · + an−1X n−1 + X n

6.3.4. Proposition. Let I ⊂ R be an ideal and M a finite module. If IM = M

then I + Ann(M) = R. That is there is a ∈ I such that (1 + a)M = 0.

Pro**of**. By 6.3.3

Put a = a0 + · · · + an−1.

a01M + · · · + 1M = (a0 + · · · + an−1)1M + 1M = 0

6.3.5. Corollary. Let I ⊂ R be an ideal and M a finite module. If IM = M and

all elements 1 + I are nonzero divisors on M, then M = 0.

6.3.6. Corollary. Let I ⊂ R be an ideal and N ⊂ M a submodule. Suppose M/N

is a finite module and M = N + IM. Then I + (N : M) = R.

6.3.7. Proposition. Let R be a ring and M a finite module. If a homomorphism

f : M → M is surjective, then it is an isomorphism.

Pro**of**. Regard M, f as a module over R[X] 6.3.1. Then (X)M = M, so by 6.3.4

there is p ∈ R[X] such that 1 + pX ∈ Ann(M). For any u ∈ Ker f, calculate

u = u + p(f) ◦ f(u) = (1 + pX)u = 0. So f is an isomorphism.

6.3.8. Exercise. (1) Let √ 0M = M. Show that M = 0.

6.4. Nakayama’s Lemma

6.4.1. Proposition. Let (R, P ) be a local ring and M a finite module. The following

conditions are equivalent.

(1) M = 0.

(2) P M = M.

(3) M ⊗R k(P ) = 0.

Pro**of**. (1) ⇒ (2) ⇔ (3) is clear. (2) ⇒ (1): Elements in 1 + P are units in R, so

by 6.3.5 M = 0.

6.4.2. Corollary. Let (R, P ) be a local ring and N ⊂ M a submodule. Suppose

M/N is a finite module and M = N + P M. Then N = M.

6.4.3. Corollary. Let (R, P ) be a local ring and M, N finite modules. If M ⊗R

N = 0, then M = 0 or N = 0.

Pro**of**. If M, N = 0 then M ⊗R k(P ), N ⊗R k(P ) = 0 are vector spaces over

k(P ). Now M ⊗R N ⊗R k(P ) M ⊗R k(P )⊗ k(P ), N ⊗R k(P ) = 0, giving the

statement.

6.4.4. Corollary. Let R be a ring and M a finite module. The following conditions

are equivalent.

(1) M = 0.

(2) P MP = MP for all prime ideals P .

(3) P MP = MP for all maximal ideals P .

(4) M ⊗R k(P ) = 0 for all prime ideals P .

(5) M ⊗R k(P ) = 0 for all maximal ideals P .

Pro**of**. Combine 6.4.1 with 5.4.1.

6.4. NAKAYAMA’S LEMMA 79

6.4.5. Corollary. Let (R, P ) be a local ring and f : M → N a homomorphism.

Assume N is finite. The following conditions are equivalent.

(1) f is surjective.

(2) f(P ) is surjective.

Pro**of**. f is surjective if and only if Cok f = 0. Cok f is finite, so it is zero if and

only if Cok f ⊗R k(P ) = Cok(f(P )) = 0, 6.4.1.

6.4.6. Corollary. Let R be a ring and f : M → N a homomorphism. Assume N

is finite. The following conditions are equivalent.

(1) f is surjective.

(2) f(P ) is surjective for all prime ideals P .

(3) f(P ) is surjective for all maximal ideals P .

6.4.7. Corollary. Let (R, P ) be a local ring and M a finite module. Let x1, . . . , xn ∈

M. The following conditions are equivalent.

(1) x1, . . . , xn generate M.

(2) x1, . . . , xn generate M ⊗R k(P ).

6.4.8. Corollary. Let R be a ring and M a finite module. Let x1, . . . , xn ∈ M.

The following conditions are equivalent.

(1) x1, . . . , xn generate M.

(2) x1, . . . , xn generate M ⊗R k(P ) for all prime ideals P .

(3) x1, . . . , xn generate M ⊗R k(P ) for all maximal ideals P .

6.4.9. Proposition. Let f : R n → R m be a homomorphism represented by an

m × n-matrix A. The following are equivalent.

(1) f is surjective.

(2) n ≥ m and the ideal (m − minors) = R.

Pro**of**. (1) ⇒ (2): If f is surjective, then for any maximal ideal f(P ) is surjective

linear map. So n ≥ m and some m-minor is nonzero in k(P ). Therefore (m −

minors) is not contained in P , so (m − minors) = R. (2) ⇒ (1): f(P ) is

surjective for all maximal ideals, so f surjective by 6.4.6.

6.4.10. Proposition. Let f : R n → R m be a homomorphism represented by an

m × n-matrix A. The following are equivalent.

(1) f is injective.

(2) n ≤ m and the ideal Ann(n − minors) = 0.

Pro**of**. See the pro**of** 6.2.5 (2).

6.4.11. Exercise. (1) Let R be a domain and f : R n → R m a homomorphism represented

by an m × n-matrix. Show that f is injective if and only if n ≤ m and some

n − minor = 0.

80 6. FINITE MODULES

6.5. Finite Presented Modules

6.5.1. Definition. Let R be a ring. A finite presented module is a module M

having an exact sequence

R n → R m → M → 0

or equivalently there is a short exact sequence

with N finite.

0 → N → R m → M → 0

6.5.2. Example. A finite projective module is finite presented, 6.1.10.

6.5.3. Lemma. For a short exact sequence

0

f

M

g

N

the following h**old**:

(1) If M, L are finite presented, then N is finite presented.

(2) If L is finite presented and N is finite , then M is finite.

(3) If N is finite presented and M is finite , then L is finite presented.

Pro**of**. (1) Choose u : R n → M → 0 and v : R m → L → 0 exact with finite

kernels. By 3.5.4 there is w : R m → N such that g ◦ w = v. There is a diagram

0

0

Rn

u

M

f

L

Rn ⊕ Rm

f◦u+w

g

N

0

Rm

v

L

By the snake lemma 3.2.4 the sequence 0 → Ker u → Ker f ◦u+w → Ker v → 0

is exact. By 6.1.5 Ker f ◦ u + w is finite. (2) Choose v : R m → L → 0 exact with

finite kernel and w : R m → N such that g ◦ w = v. There is a diagram

0

0

0

f

M

R m

w

N

g

Rm

v

L

By the snake lemma 3.2.4 the sequence 0 → Ker w → Ker v → M → Cok w →

0 is exact. By 6.1.5 M is finite. (3) Choose w : R m → N → 0 exact with finite

kernel. Then v = g ◦ w : R m → L → 0 is exact and there is a diagram

0

0

0

f

M

R m

w

N

g

Rm

v

L

By the snake lemma 3.2.4 the sequence 0 → Ker w → Ker v → M → 0 is exact.

By 6.1.5 Ker v is finite and L is finite presented.

6.5.4. Corollary. Let

0

f

M

g

N

be a split exact sequence. Then N is finite presented if and only if M, L are finite

presented.

L

0

0

0

0

0

0

0

Pro**of**. By 3.1.13 there is a split exact sequence

so the statement follows from 6.5.3.

0

6.5. FINITE PRESENTED MODULES 81

v

L

u

N

M

6.5.5. Corollary. Let f : M → N be a homomorphism.

(1) If M is finite and Im f finite presented, then Ker f is finite.

(2) If Ker f, Im f are finite presented, then M is finite presented.

(3) If N is finite presented and Im f finite, then Cok f is finite presented.

(4) If Im f, Cok f are finite presented, then N is finite presented.

Pro**of**. Use the sequences 3.1.8.

6.5.6. Proposition. Let R be a ring and M, N finite presented modules.

(1) M ⊕ N is finite presented.

(2) M ⊗R N is finite presented.

Pro**of**. (1) This is clear from 6.5.4. (2) If M = R n then M ⊗R N is finite presented

by (1). In general chose u : R n → M → 0 exact with finite kernel. The sequence

Ker u ⊗R N → R n ⊗R N → M ⊗R N → 0 is exact. So Ker u ⊗ 1N is finite.

Conclusion by 6.5.5.

6.5.7. Proposition. Given submodules N, L ⊂ M. Then

(1) If M/N, M/L are finite and M/N + L is finite presented, then M/N ∩ L is

finite.

(2) If M/N, M/L are finite presented and M/N ∩ L is finite, then M/N + L is

finite presented.

Pro**of**. Use the sequence 3.2.7.

6.5.8. Proposition. Let R be a ring and U a multiplicative subset.

(1) For a finite module M and any module N the natural homomorphism

0

U −1 HomR(M, N) → Hom U −1 R(U −1 M, U −1 N)

is injective.

(2) For a finite presented module M and any module N the homomorphism

U −1 HomR(M, N) → Hom U −1 R(U −1 M, U −1 N)

is a natural isomorphism.

Pro**of**. (1) If M = R this is an isomorphism. Then this is also an isomorphism

for M = R n since both functors respect finite direct sums. In general choose

0 → K → R n → M → 0 exact. There is a diagram

0

0

U −1 HomR(M, N)

HomU −1R(U −1M, U −1N)

U −1 HomR(R n , N)

HomU −1R(U −1Rn , U −1N) giving injectivity. (2) In (1) K is finite, so the last vertical map is injective. Conclusion

by the five lemma 3.2.8.

U −1 HomR(K, N)

HomU −1R(U −1K, U −1N)

82 6. FINITE MODULES

6.5.9. Corollary. Let P ⊂ R be a prime ideal and M a finite presented module.

For any module N there is a natural isomorphism

HomR(M, N)P HomRP (MP , NP )

6.5.10. Proposition. Let R be a ring and F a module. The following conditions

are equivalent.

(1) F is flat.

(2) For any module N and a relation 0 =

i yi ⊗ xi ∈ N ⊗R F , there exist

zj ∈ F and aij ∈ R such that 0 =

i aijyi ∈ N and xi =

j aijzj ∈ F .

(3) For any relation 0 =

i bixi ∈ F , there exist zj ∈ F and aij ∈ R such that

0 =

i aijbi ∈ R and xi =

j aijzj ∈ F .

Pro**of**. (1) ⇒ (2): Let f : Rn → N, ei ↦→ yj, then 0 → Ker f ⊗R F → F n →

N ⊗R F is exact. By assumption (xi) ∈ Ker f ⊗R F , so (xi) =

j aij ⊗ zj

with (aij) ∈ Ker f. (2) ⇒ (3) is clear. (3) ⇒ (1): Let I ⊂ R be an ideal and

bi ⊗ xi ∈ Ker(I ⊗R F → F ). Then 0 =

i aijbi and xi =

j aijzj. Now

calculate bi ⊗ xi =

j i aijbi ⊗ xi = 0. By 3.7.12 F is flat.

6.5.11. Proposition. Let (R, P ) be a local ring and F a finite presented module.

The following conditions are equivalent.

(1) F is free.

(2) F is projective.

(3) F is flat.

(4) P ⊗R F → F is injective.

Pro**of**. (1) ⇒ (2) ⇒ (3) ⇒ (4) are clear. (4) ⇒ (1): Choose xi ∈ F such that

xi ⊗ 1 give a basis for F ⊗R k(P ). The homomorphism f : R n → F, ei ↦→ xi is

surjective by 6.4.5. P ⊗R Ker f → P n → P ⊗R F → 0 is exact, so P Ker f =

Ker(P n → P ⊗R F ) = Ker(P n → P ⊗R F → F ) = Ker f. by the hypothesis.

Ker f is finite 6.5.5 and therefore Ker f = 0 by 6.4.1, so F is free.

6.5.12. Corollary. Let (R, P ) be a local ring and f : F → F ′ a homomorphism

**of** finite free modules. The following conditions are equivalent.

(1) f has a retraction u : F ′ → F .

(2) f is injective and Cok f is free.

(3) f(P ) is injective.

Pro**of**. (1) ⇒ (2): Cok f is projective, so free by 6.5.11. (2) ⇒ (3) is clear. (3) ⇒

(1): Let f ∨ : F ′∨ → F ∨ be the dual homomorphism. f ∨ (P ) is surjective, so f ∨ is

surjective, 6.4.5. A section v **of** f ∨ gives a retraction u = v ∨ .

6.5.13. Corollary. Let R be a ring and F a finite presented module. The following

conditions are equivalent.

(1) F is projective.

(2) F is flat.

(3) F is locally free.

(4) FP is free for all maximal ideals P .

Pro**of**. (1) ⇒ (2) ⇒ (3) ⇒ (4) are clear by 6.5.11. (4) ⇒ (1): Let N → L → 0

be exact. By hypothesis HomRP (FP , NP ) → HomRP (FP , LP ) → 0 is exact for

all maximal ideals. By 6.5.9 HomR(F, N)P → HomR(F, L)P → 0 is exact for

6.6. FINITE RING HOMOMORPHISMS 83

all maximal ideals. By 5.4.3 HomR(F, N) → HomR(F, L) → 0 is exact and F is

projective.

6.5.14. Exercise. (1) Let I ⊂ R be an ideal. Show that R/I is a finite presented

R-module if and only if I is a finite ideal.

(2) Show that Q is a flat, but not projective Z-module.

6.6. Finite ring homomorphisms

6.6.1. Definition. A ring homomorphism φ : R → S is a finite ring homomorphism

if S is a finite R-module. If R ⊂ S is a subring, then a finite ring homomorphism

is a finite ring extension.

6.6.2. Proposition. Let R be a ring.

(1) Let f ∈ R[X] be a monic polynomial. Then the homomorphism R →

R[X]/(f) is finite.

(2) Let f : M → M be a homomorphism **of** a finite R-module. Then the homomorphism

6.3.1, R → R[f] is finite.

Pro**of**. (2) Follows from (1) and 6.3.3.

6.6.3. Lemma. Let φ : R → S be a finite ring homomorphism. If N is a finite

S-module, then by restriction along φ the R-module N is finite.

6.6.4. Proposition. Let R ⊂ S be a finite ring extension **of** domains. Then R is a

field if and only if S is a field.

Pro**of**. Let R be a field, a minimal equation 6.3.3 for scalar multiplication by a

nonzero b ∈ S, bS as R-module homomorphism

gives

b n + · · · + a0 = 0

b −1 = −a −1

0 (an−1b n−2 + · · · + a1) ∈ S

Let S be a field and 0 = a ∈ R. An equation 6.3.3 for scalar multiplication a −1

S as

R-homomorphism

a −n + · · · + a0 = 0

gives

a −1 = −(a0a n−1 + · · · + an−1) ∈ R

6.6.5. Corollary. Let R → S be a finite ring homomorphism. A prime ideal Q ⊂ S

is maximal if and only if the contraction Q ∩ R is maximal.

6.6.6. Proposition (going-up). Let R ⊂ S be a finite ring extension and P ⊂ R a

prime ideal. Then there is a prime ideal Q ⊂ S contracting

.

P = Q ∩ R

Pro**of**. RP ⊂ SP is a finite ring extension. Since SP = 0 there is a maximal

ideal in SP contracting to P RP by 6.6.5. The corresponding prime ideal Q ⊂ S

contracts to P .

6.6.7. Proposition. Let R ⊂ S be a finite ring extension and E an R-module. The

following are equivalent.

84 6. FINITE MODULES

(1) E is an injective R-module.

(2) HomR(S, E) is an injective S-module.

Pro**of**. (1) ⇒ (2): This is 3.6.10. (2) ⇒ (1): Let E ⊂ E ′ be an injective envelope,

3.6.15. HomR(S, E) → HomR(S, E ′ ) is injective and identifies HomR(S, E)

as an essential submodule, since S is a finite R-module. E ′ is injective, so also

HomR(S, E ′ ). It follows that

HomR(S, E)

HomR(R, E)

E

HomR(S, E ′ )

HomR(R, E ′ )

E ′

with the right down map being surjective. So E = E ′ .

6.6.8. Exercise. (1) Show that Z → Z[ √ −5] is finite.

(2) Let p be a prime number. Show that Z → Z (p) is not finite.

7

Modules **of** finite length

7.1. Simple Modules

7.1.1. Definition. A nonzero module M is a simple module if 0 and M are the

only submodules.

7.1.2. Proposition. Let f : M → M ′ be a homomorphism

(1) If M is simple, then f is either zero or injective.

(2) If M ′ is simple, then f is either zero or surjective.

(3) If M, M ′ are both simple, then f is either zero or an isomorphism.

Pro**of**. This follows from 2.3.3.

7.1.3. Proposition. A simple module is isomorphic to a factor module R/P for

some uniquely determined maximal ideal P . A simple module is finite, in fact

generated by one element.

Pro**of**. A nonzero f : R → M must be surjective and R/P is simple exactly when

P is maximal. Clearly P = Ann(M).

7.1.4. Corollary. Any nonzero finite module has a simple factor module.

Pro**of**. If M = 0 then MP = 0 for some maximal ideal. By 6.4.1 M ⊗R k(P ) =

M/P M = 0. Choose a nonzero linear projection M/P M → k(P ), giving M →

k(P ) → 0 exact.

7.1.5. Example. (1) If K is a field, a one dimensional vector space is simple.

(2) If R is a principal ideal domain, then R/(p) is a simple module for all irreducible

(p).

(3) Z/(p) are simple for all prime numbers p.

(4) Let K be a field. K[X]/(X − a) is a simple K[X]-module.

7.1.6. Lemma. Let I ⊂ R be an ideal. A simple R/I-module is a simple Rmodule.

7.1.7. Exercise. (1) Show that Z does not contain any simple modules.

(2) Show that Q does not have any simple factor Z-module.

(3) Let L, L ′ ⊂ M be simple submodules. Show that either L ∩ L ′ = 0 **of** L = L ′ .

(4) Let L = L ′ ⊂ M be simple submodules. Show that L ⊕ L ′ = L + L ′ .

7.2. The Length

7.2.1. Definition. Let R be a ring. A module M is **of** finite length if it admits a

composition series by submodules

0 = M0 ⊂ M1 ⊂ · · · ⊂ Mn−1 ⊂ Mn = M

such that each factor Mi/Mi−1 is a simple module.

85

86 7. MODULES OF FINITE LENGTH

7.2.2. Lemma. Any two finite composition series have the same number **of** submodules.

Pro**of**. Let l(M) be the least length **of** a composition series. If M is simple then

l(M) = 1. Let 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mn = M be any composition series. For a

submodule N ⊂ M there is a filtration 0 = N ∩M0 ⊂ N ∩M1 ⊂ · · · ⊂ N ∩Mn =

N with factors N ∩Mi/N ∩Mi−1 ⊂ Mi/Mi−1 being either simple or 0. It follows

that l(N) ≤ l(M). If l(N) = l(M) then each factor is nonzero and therefore

N = M. Applying this to the composition series **of** M gives l(M) ≤ n ≤ l(M)

as wanted.

7.2.3. Definition. The number **of** nontrivial submodules in a filtration as above

will be denoted ℓR(M) and is the length **of** M.

7.2.4. Proposition. Given an exact sequence

0 → N → M → L → 0

Then M has a finite length if and only if both N and L have finite length. In that

case

ℓR(M) = ℓR(N) + ℓR(L)

Pro**of**. A finite filtration with simple quotients in M induces filtrations in both N

and L, and vice versa.

The filtrations in N and L give a filtration in M with the same quotients, so the

length formula follows from 7.2.2.

7.2.5. Corollary. Let f : M → N be a homomorphism.

(1) If M has finite length if and only if Ker f, Im f have finite length. If finite

length

ℓR(M) = ℓR(Ker f) + ℓR(Im f)

(2) If N has finite length if and only if Im f, Cok f have finite length. If finite

length

ℓR(N) = ℓR(Im f) + ℓR(Cok f)

Pro**of**. Use the sequences 3.1.8.

7.2.6. Corollary. Let f : M → M be a homomorphism on a module **of** finite

length. Then

ℓR(Ker f) = ℓR(Cok f)

and the following are equivalent:

(1) f is injective.

(2) f is surjective.

(3) f is an isomorphism.

Pro**of**. Use the sequences 3.1.8.

7.2.7. Corollary. Let N ⊂ M be a submodule and suppose M has finite length.

(1) ℓR(N) ≤ ℓR(M).

(2) ℓR(N) = ℓR(M) if and only if N = M.

7.2.8. Corollary. Let N, L ⊂ M be a submodules and suppose N, L has finite

length. Then N + L, N ∩ L have finite length and

ℓR(N + L) + ℓR(N ∩ L) = ℓR(N) + ℓR(L)

7.3. ARTINIAN RINGS 87

7.2.9. Proposition. Given submodules N, L ⊂ M. Then the following are equivalent.

(1) M/N, M/L have finite length.

(2) M/N ∩ L has finite length.

If finite length

ℓR(M/N + L) + ℓR(M/N ∩ L) = ℓR(M/N) + ℓR(M/L)

Pro**of**. Use the sequences 3.2.8.

7.2.10. Proposition. A R-module M **of** finite length is finite and generated by

ℓR(M) or less elements.

Pro**of**. There is a sequence 0 → N → M → L → 0 with L simple. Conclusion

by induction.

7.2.11. Proposition. Let I ⊂ R be an ideal. Suppose an R/I-module M has finite

length. Then M has finite length as R-module and

ℓ R/I(M) = ℓR(M)

Pro**of**. This follows from 7.1.6 and 7.2.2.

7.2.12. Example. Let K be a field. A module M is **of** finite length if it is a finite

dimensional vector space. Then

7.2.13. Exercise. (1) Compute

(2) Compute

ℓZ(Z/(p n1

1

ℓK(M) = rankK M

. . . pnk

k )) = n1 + · · · + nk

ℓ K[X](K[X]/((X − a1) n1 . . . (X − ak) nk )) = n1 + · · · + nk

7.3. Artinian Rings

7.3.1. Lemma. Let M be a module. The following conditions are equivalent.

(1) Any decreasing sequence Mi **of** submodules is stationary, Mn = Mn+1 for

n >> 0.

(2) Any nonempty subset **of** submodules **of** M contains a minimal element.

Pro**of**. (1) ⇒ (2): Suppose a nonempty subset **of** submodules do not contain a

minimal element. Then choose a non stationary sequence. (2) ⇒ (1): A descending

sequence containing a minimal element is stationary.

7.3.2. Definition. A module M which satisfies the conditions **of** 7.3.1 is an artinian

module.

7.3.3. Definition. A ring R is an artinian ring if R is an artinian module over

itself.

7.3.4. Proposition. Let 0 → N → M → L → 0 be an exact sequence **of** modules

over the ring R. Then M is artinian if and only if N and L are artinian.

88 7. MODULES OF FINITE LENGTH

Pro**of**. Let M be artinian. A chain in N gives a chain in M, so N is artinian. A

chain in L gives a chain in M, which becomes stationary, so also the original chain,

so L is artinian. Conversely if N and L are artinian and Mi a chain in M, then

the induced chains in N and L become stationary. By the snake lemma 3.2.4 the

original chain is stationary.

7.3.5. Corollary. Let f : M → N be a homomorphism.

(1) M is artinian if and only if Ker f, Im f are artinian.

(2) N is artinian if and only if Im f, Cok f are artinian

Pro**of**. Use the sequences 3.1.8.

7.3.6. Proposition. Let f : M → M be a homomorphism on an artinian module.

Then the following are equivalent

(1) f is injective

(2) f is an isomorphism

Pro**of**. There is a number n such that Im f ◦n = Im f ◦2n . For x ∈ M there is y

such that f ◦n (x) = f ◦2n (y). Then f ◦n (x − f ◦n (y)) = 0 so x = f ◦n (y) since f is

injective. It follows that f ◦n is surjective and so is f.

7.3.7. Proposition. Given submodules N, L ⊂ M. Then the following are equivalent.

(1) M/N, M/L are artinian

(2) M/N ∩ L is artinian

Pro**of**. use the sequences 3.2.8.

7.3.8. Corollary. Given ideals I, J ⊂ R. Then the following are equivalent.

(1) R/I, R/J are artinian

(2) R/I ∩ J is artinian

7.3.9. Corollary. Let R be an artinian ring and I an ideal. Then the factor ring

R/I is artinian.

The product **of** two artinian rings is artinian.

7.3.10. Proposition. An artinian domain is a field.

Pro**of**. By 7.3.6 scalar multiplication with a nonzero element is an isomorphism.

7.3.11. Proposition. Let R be an artinian ring. Then all prime ideals are maximal

and there are only finitely many such.

Pro**of**. By 7.3.10 primes are maximal. If Pi are different maximal ideals, then

P1 · · · Pn+1 ⊂ P1 · · · Pn is a strictly decreasing chain. So there are only finitely

many maximal ideals.

7.3.12. Proposition. Let R be an artinian ring.

(1) The factor ring R/ √ 0 is a finite product **of** fields.

(2) The nilradical √ 0 is nilpotent, √ 0 k = 0 for some k.

7.3. ARTINIAN RINGS 89

Pro**of**. (1) Let P1, . . . , Pn be the prime and maximal ideals 7.3.11. The nilradical

√ 0 = P1 ∩ · · · ∩ Pn by 5.1.7. Conclude by Chinese remainders 1.4.3. (2) √ 0 k =

√ 0 k+1 for some k. If √ 0 k = 0 let (a) be minimal among ideals I such that

I √ 0 k = 0. By minimality (a) = (a) √ 0 k , so a = ab for some b ∈ √ 0 k . But b is

nilpotent 1.3.8, so a = 0 gives a contradiction. It follows, that √ 0 k = 0.

7.3.13. Proposition. A ring R is artinian if and only if it has finite length.

Pro**of**. The factor module √ 0 i / √ 0 i+1 is an artinian module over R/ √ 0 which is

a product **of** fields, so it has finite length.

7.3.14. Corollary. Let R be an artinian ring and M a module. The following are

equivalent.

(1) M is finite.

(2) M has finite length.

(3) M is finite presented.

7.3.15. Corollary. Let R be artinian and R → S a finite ring homomorphism.

(1) S is artinian.

(2) A finite length S-module N is by restriction **of** scalars a finite length Rmodule.

(3) A finite length R-module M gives by change **of** rings M ⊗R S as finite length

S-module.

7.3.16. Proposition. Let M be a R-module **of** finite length.

(1) The ring R/ Ann(M) is artinian.

(2) There are only finitely many prime ideals Ann(M) ⊂ P .

(3) Any prime ideal Ann(M) ⊂ P is maximal.

(4) M is finite presented.

Pro**of**. (1) Let x ∈ M, then R/ Ann(x) Rx is artinian. If x1, . . . , xn generate

M, then Ann(M) = Ann(x1)∩· · ·∩Ann(Xn). By 7.3.8 R/ Ann(M) is artinian.

7.3.17. Example. Let K be a field. The ring R =

artinian.

(1) The maximal ideals in R are

(2) The simple types are

N

Pi = {a : N → K|a(i) = 0}

R/Pi Ji = {a : N → K|a(j) = 0, i = j}

(3) Ji are the only simple ideals and the sum is an ideal

Ji = J = R

(4) J is an ideal which has no complement in R.

i

K = {a : N → K} is not

7.3.18. Exercise. (1) Show that a vector space is artinian if and only if it is finite dimensional.

(2) Show that Z is not artinian.

(3) Show that R[X] is not artinian for a nonzero ring R.

(4) Show that Q[X]/(X 2 − X) is artinian.

90 7. MODULES OF FINITE LENGTH

(5) Show that a ring with finitely many ideals is artinian.

(6) Let R be a principal ideal domain and a = 0. Show that R/(a) is artinian.

7.4. Localization

7.4.1. Proposition. Any artinian ring is the product **of** finitely many artinian local

rings. Let R be artinian with maximal ideals P1, . . . , Pk. Then there is n1, . . . , nk

such that P n1

1 . . . P nk

k = 0 and

R R/P n1

nk

1 × · · · × R/Pk Each R/P ni

1 is a local artinian ring.

Pro**of**. This follows from 7.3.11, 7.3.12 and Chinese remainders 1.4.2.

7.4.2. Example. A reduced artinian ring is a finite product **of** fields.

7.4.3. Corollary. Let R be an artinian ring and U ⊂ R a multiplicative subset.

The ring **of** fractions U −1 R is an artinian ring.

7.4.4. Example. Let P ⊂ R be a maximal ideal and M a module. Assume s /∈ P

and n ∈ N.

(1) Rs + P n = A.

(2) Scalar multiplication by s : M/P n M → M/P n M, x → sx is an isomorphism.

(3) The canonical map M/P n M → (M/P n M)P is an isomorphism.

(4) If M is finite then M/P n M has finite length.

7.4.5. Proposition. Let R be artinian with maximal ideals P1, . . . , Pk. A finite

module M has a decomposition

M

The length is

Pro**of**. Let R R/P n1

ℓR(M) =

R/P nk

k and MPi M ⊗R R/P ni

i .

1

i

i

MPi

ℓRP i (MPi )

nk

× · · · × R/Pk . Then M M ⊗R R/P n1

1 × · · · × M ⊗R

7.4.6. Proposition. Let M be an R-module **of** finite length and Ann(M) ⊂ P1, . . . , Pk

the maximal ideals. Then MPi is a finite length RPi-module and

ℓR(M) =

ℓRP (MPi i )

Pro**of**. The ring R/ Ann(M) R/P n1

1

i

× · · · × R/P nk

k

by 7.3.16 and 7.4.1.

7.4.7. Proposition. Let (R, P ) → (S, Q) be a local ring homomorphism and assume

that k(P ) → k(Q) is a finite extension. If N is a finite length S-module, then

N is a finite length R-module and

Pro**of**. Reduce to N = k(Q).

ℓR(N) = rank k(P )(k(Q)) · ℓS(N)

7.4.8. Proposition. Let (R, P ) → (S, Q) be a local ring homomorphism and assume

that S/P S is a finite length S-module. Let M be a finite length R-module.

7.5. LOCAL ARTINIAN RING 91

(1) M ⊗R S is a finite length S-module.

(2) In general

ℓS(M ⊗R S) ≤ ℓS(S/P S) · ℓR(M)

(3) If R → S is flat then

ℓS(M ⊗R S) = ℓS(S/P S) · ℓR(M)

Pro**of**. The case M = k(P ) is clear. Conclude by induction.

7.4.9. Exercise. (1) Let K ⊂ L be a finite field extension and W a finite vector space

over L. Show that

rankK(W ) = rankK(L) · rankL(W )

(2) Let K ⊂ L be a finite field extension and V a finite vector space over K. Show that

rankL(V ⊗K L) = rankK(V )

7.5. Local artinian ring

7.5.1. Lemma. Let (R, P ) be a local artinian ring and M a finite module. The

following are equivalent.

(1) M = 0.

(2) HomR(k(P ), M) = 0.

(3) HomR(M, k(P )) = 0.

Pro**of**. Clear by a filtration.

7.5.2. Proposition. Let (R, P ) be a local artinian ring and k(P ) ⊂ E an injective

envelope.

(1) The are isomorphisms

k(P ) HomR(k(P ), k(P )) HomR(k(P ), E)

(2) For a finite module M, the module HomR(M, E) has finite length and

ℓR(HomR(M, E)) = ℓR(M)

Pro**of**. (1) A nonzero homomorphism f : k(P ) → E has Im f = k(P ) since the

extension is essential. (2) This follows from (1) by use **of** a filtration.

7.5.3. Corollary. Let (R, P ) be a local artinian ring and k(P ) ⊂ E an injective

envelope. Then E has finite length

ℓR(E) = ℓR(R)

7.5.4. Proposition. Let (R, P ) be a local artinian ring and k(P ) ⊂ E an injective

envelope. There is a natural isomorphism for any finite module M

x ↦→ evx : M → HomR(HomR(M, E), E)

Pro**of**. The case M = k(P ) is clear by 7.5.2. Let 0 → N → M → L → 0 be a

short exact sequence. Then the following diagram has exact rows.

0

0

N

HomR(HomR(N, E), E)

M

HomR(HomR(M, E), E)

Conclusion by the five lemma 3.2.8 and induction on the length.

L

HomR(HomR(L, E), E)

0

0

92 7. MODULES OF FINITE LENGTH

7.5.5. Corollary. Let (R, P ) be a local artinian ring and k(P ) ⊂ E an injective

envelope. There is an isomorphism

11E : R → HomR(E, E)

7.5.6. Proposition. Let (R, P ) be a local artinian ring and k(P ) ⊂ E an injective

envelope. Let M → E ′ be an injective envelope **of** a finite module. Then

(1) HomR(k(P ), M) HomR(k(P ), E ′ )

(2) E ′ E n , where n = ℓR(HomR(k(P ), M).

(3) ℓR(E ′ ) = ℓR(HomR(k(P ), M) ℓR(E).

Pro**of**. (1) A homomorphism f : k(P ) → E ′ has Im f ⊂ M since the extension

is essential. (2) By induction on length **of** M follows that E ′ E n , where n is

determined by (1) and 7.5.2. (3) This follows from (2).

7.5.7. Proposition. Let (R, P ) be a local artinian ring and k(P ) ⊂ E an injective

envelope. The following are equivalent.

(1) R is injective.

(2) R E.

(3) HomR(k(P ), R) HomR(k(P ), k(P )) k(P ).

Pro**of**. (1) ⇒ (2): By 7.5.6 R E n and by 7.5.3 n = 1. (2) ⇒ (3): Immediate

from 7.5.6. (3) ⇒ (1): Let R ⊂ E ′ be an injective envelope. By 7.5.6 R = E ′ .

7.5.8. Definition. A ring satisfying the conditions 7.5.7 is a local artinian Gorenstein

ring.

7.5.9. Example. Let R be a principal ideal domain and p ∈ R an irreducible

element. Then R/(p n ) is a local artinian Gorenstein ring.

7.5.10. Exercise. (1) Let p be a prime number. Show that Z/(p k ) is a local artinian

Gorenstein ring.

8

Noetherian modules

8.1. Modules and submodules

8.1.1. Lemma. Let M be a module. The following conditions are equivalent.

(1) Any increasing sequence Mn **of** submodules is stationary, Mn = Mn+1 for

n >> 0.

(2) Any nonempty subset **of** submodules **of** M contains a maximal element.

(3) Any submodule **of** M is finite.

Pro**of**. (1) ⇔ (2): See the pro**of** **of** 7.3.1. (2) ⇒ (3): Let N be a submodule **of**

M and choose a maximal element N ′ in the set **of** finite submodules **of** N. For

y ∈ N, the module N ′ + Ry is finite, so N ′ = N ′ + Ry gives N = N ′ finite.

(3) ⇒ (1): The union ∪Mn is a submodule, generated by x1, . . . , xm ∈ Mk, so

Mn = ∪Mn, n > k.

8.1.2. Definition. A module M which satisfies the conditions **of** 8.1.1 is a noetherian

module.

8.1.3. Proposition. Let 0 → N → M → L → 0 be an exact sequence **of** modules

over the ring R. Then M is noetherian if and only if N and L are noetherian.

Pro**of**. See the pro**of** **of** 7.3.4.

8.1.4. Corollary. Let f : M → N be a homomorphism.

(1) M is noetherian if and only if Ker f, Im f are noetherian.

(2) N is noetherian if and only if Im f, Cok f are noetherian.

Pro**of**. Use the sequences 3.1.8.

8.1.5. Proposition. Let f : M → M be a homomorphism on a noetherian module.

Then the following are equivalent

(1) f is surjective

(2) f is an isomorphism

Pro**of**. Analog **of** the pro**of** **of** 7.3.4. There is a number n such that Ker f ◦2n =

Ker f ◦n . For x ∈ Ker f ◦n there is y such that x = f ◦n (y). Then f ◦2n (y) =

f ◦n (x) = 0, so y ∈ Ker f ◦2n = Ker f ◦n . Then x = f ◦n (y) = 0 and f is

injective.

8.1.6. Proposition. A finite direct sum **of** noetherian modules is noetherian.

8.1.7. Proposition. Given submodules N, L ⊂ M. Then the following are equivalent.

(1) M/N, M/L are noetherian

(2) M/N ∩ L is noetherian

93

94 8. NOETHERIAN MODULES

Pro**of**. Use the sequence 3.2.7.

8.1.8. Example. A module **of** finite length is noetherian.

8.1.9. Exercise. (1) Show that

N Z is not noetherian.

(2) Show that

N Z is not noetherian.

(3) Show that Q is not a noetherian Z-module.

8.2. Noetherian rings

8.2.1. Definition. A ring R is a noetherian ring if R is a noetherian module.

8.2.2. Proposition. The following conditions are equivalent.

(1) R is noetherian.

(2) Any ideal is finite.

(3) Any increasing sequence **of** ideals is stationary.

(4) Any nonempty subset **of** ideals **of** contains an ideal maximal for inclusion.

Pro**of**. This is follows from 8.1.1.

8.2.3. Proposition. The nilradical in a noetherian ring is nilpotent, i.e. √ 0 n = 0

for some n. Some power **of** the radical **of** an ideal is contained in the ideal, i.e.

√ I n ⊂ I for some n

Pro**of**. √ 0 = (b1, . . . , bn) such that b k i = 0. Then ( aibi) nk = 0.

8.2.4. Proposition. Given ideals I, J ⊂ R. Then the following are equivalent.

(1) R/I, R/J are noetherian

(2) R/I ∩ J is noetherian

Pro**of**. This is a special case **of** 8.1.7.

8.2.5. Lemma. (1) Any factor ring **of** a noetherian ring is a noetherian ring.

(2) A principal ideal domain is noetherian.

(3) If M is a noetherian R-module, then R/ Ann(M) is a noetherian ring.

Pro**of**. (1), (2) are clear. (3) Analog **of** the pro**of** **of** 7.3.16. Let x ∈ M, then

R/ Ann(x) Rx is noetherian. If x1, . . . , xn generate M, then Ann(M) =

Ann(x1) ∩ · · · ∩ Ann(Xn). By 8.2.4 R/ Ann(M) is artinian.

8.2.6. Proposition. Let R be a noetherian ring. Any finite R-module is noetherian.

Pro**of**. Let M be a finite R-module. There is a surjective homomorphism R n →

M → 0, 6.1.2. Conclusion by 8.2.3.

8.2.7. Corollary. Let R be a noetherian ring. Any finite R-module is finite presented.

8.2.8. Lemma. Let R be a noetherian ring and M, N noetherian modules.

(1) M ⊗R N is noetherian.

(2) HomR(M, N) is noetherian.

Pro**of**. Conclusion by 6.1.8, 6.1.9, 8.2.6.

8.3. FINITE TYPE RINGS 95

8.2.9. Proposition. Let R be a noetherian ring and U a multiplicative subset. For

a finite module M and any module N the homomorphism

is an isomorphism.

U −1 HomR(M, N) → Hom U −1 R(U −1 M, U −1 N)

Pro**of**. Conclusion by 6.5.8, 8.2.7.

8.2.10. Proposition. Let R be a noetherian ring. If E is an injective R-module,

then U −1 E is an injective U −1 R-module.

Pro**of**. Let I ⊂ R be an ideal. Then HomR(R, E) → HomR(I, E) → 0 is exact.

By 8.2.9 Hom U −1 R(U −1 R, U −1 E) → Hom U −1 R(U −1 I, U −1 E) → 0 is exact.

So U −1 E is injective 3.6.7, using that any ideal is extended 4.3.6.

8.2.11. Proposition. For a ring R, the following are equivalent.

(1) R is a noetherian ring.

(2) For any family Eα **of** injective modules, the sum

α Eα is injective.

Pro**of**. (1) ⇒ (2): Let I ⊂ R be an ideal. A homomorphism f : I → Eα

has Im f contained in a finite sum, which is injective 3.6.6. So f extends to R →

Eα, giving injectivity. (2) ⇒ (1): Let In ⊂ R be an increasing chain **of** ideals.

Put I = ∪In and choose an injective envelope I/In ⊂ En. The homomorphism

f : I → En extends to f ′ : R → En. Since Im f is contained in a finite

sum, I/In = 0 for n >> 0.

8.2.12. Proposition. If R is a ring such that every prime ideal is finite, then it is

noetherian.

Pro**of**. If R is not noetherian, then the set **of** not finite ideals is nonempty and by

Zorn’s lemma it has a maximal element I. Since I is not prime there is a, b /∈ I

and ab ∈ I. The ideals I + (a) and I : (a) are both greater that I and therefore

finite. If I + (a) = (a1, . . . , am, a), ai ∈ I and I : (a) = (b1, . . . , bn), then

I = (a1, . . . , am, ab1, . . . , abn) is finite. It follows that R must be noetherian.

8.2.13. Exercise. (1) Show that a principal ideal domain is noetherian.

(2) Let I ⊂ R be an ideal in a noetherian ring. Show that R/I is noetherian.

(3) Let K be a field and R = K[X1, X2, . . . ] the polynomial ring in countable many

variables. Show that R is not noetherian.

(4) Let K be a field. Show that the ring R =

N K is not noetherian.

8.3. Finite type rings

8.3.1. Proposition (Hilbert’s basis theorem). Let R be a noetherian ring. Then the

ring **of** polynomials R[X] is noetherian.

Pro**of**. Assume I ⊂ R[X] to be a not finite ideal. Choose a sequence f1, f2, · · · ∈

I such that

fi = aiX di + terms **of** lower degree , ai = 0

and fi+1 has lowest degree in I − (f1, . . . , fi). The ideal **of** leading coefficients is

finitely generated by a1, . . . , an. Then an+1 = b1a1 + · · · + bnan and d1 ≤ · · · ≤

dn+1 = d gives

fn+1 − b1X d−d1 f1 − · · · − bnX d−dn fn

in I − (f1 . . . , fn) **of** degree less than d. By contradiction the ideal I is finite.

96 8. NOETHERIAN MODULES

8.3.2. Corollary. Let R be a noetherian ring and R → S a finite type ring over R.

Then S is noetherian.

8.3.3. Corollary. Let R be a noetherian ring and M a finite module. Then the ring

R ⊕ M, 2.1.14, is noetherian.

8.3.4. Example. Let I ⊂ R be an ideal in a noetherian ring. Then there are noetherian

rings.

(1) GIR = ⊕n≥0I n /I n+1 .

(2) BIR = ⊕n≥0I n .

8.3.5. Theorem (Krull’s intersection theorem). Let I be an ideal in a noetherian

ring R. Then there is a ∈ I such that

1 + a ∈ Ann(

I n )

Pro**of**. Let I = (u1, . . . , um). If b ∈ I n then b = fn(u1, . . . , um) where fn ∈

R[X1, . . . , Xm] are homogeneous **of** degree n. By Hilbert’s basis theorem, 8.3.1

there is N such that

fN+1 = f1g1 + · · · + fNgN

where gn is homogeneous **of** degree N − n + 1 > 0. By substitution b ∈ bI and

I ∩ I n = ∩I n . Conclusion by 6.3.4.

8.3.6. Corollary. Let I be an ideal in a noetherian ring R such that the elements

1 + a, a ∈ I are nonzero divisors. Then

I n = 0

n

8.3.7. Corollary. Let I be an ideal in a noetherian ring R and M a finite module.

Then there is a ∈ I such that

1 + a ∈ Ann(

I n M)

Pro**of**. Use 8.3.5 on the ring R ⊕ M and the ideal I ⊕ M. Clearly (I + M) n =

I n + I n−1 M.

8.3.8. Corollary. Let I be an ideal in a noetherian ring R and M a finite module

such that the elements 1 + a, a ∈ I are nonzero divisors on M. Then

I n M = 0

n

8.3.9. Proposition. If R ⊂ S be a finite extension. Then R is noetherian if and

only if S is noetherian.

Pro**of**. Assume S is noetherian. Let Eα be a family **of** injective R-modules. Then

HomR(S, Eα) is an injective S-module. Since S is finite over R, HomR(S, Eα)

HomR(S, Eα) 6.1.14. By 6.6.7 Eα is injective and by 8.2.11 R is noetherian.

8.3.10. Example. Let K be a field and R = K[X1, X2, . . . ]/(X1 − x2X3, x2 −

x3X4, . . . ). Put P = (X1, X2, . . . ).

(1) P is maximal

(2) P 2 = P .

(3) P (∩nP n ) = ∩nP n .

n

n

8.4. POWER SERIES RINGS 97

8.3.11. Exercise. (1) Show that if R[X] is noetherian, then R is noetherian.

(2) Show that the subring Z[2X, 2X 2 , . . . ] ⊂ Z[X] is not noetherian. Conclude that the

extension is not finite.

8.4. Power series rings

8.4.1. Proposition. Let R be a noetherian ring. Then the power series ring R[[X]]

is noetherian.

Pro**of**. Let P ⊂ R[[X]] be a prime ideal. Then P +(X)/(X) = (a1, . . . , an) ⊂ R

is a finite ideal. If X ∈ P then P = (a1, . . . , an, X) is finite. Suppose X /∈ P

and choose fi = ai + terms **of** positive degree ∈ P . If g ∈ P then Xg1 =

g − b11f1 + · · · + bn1fn ∈ P ∩ (X) for some bi1 ∈ R. Since P is prime, g1 ∈ P .

Now Xg2 = g1 − b12f1 + · · · + bn2fn ∈ P ∩ (X) and so on. Put hi = bikX k ,

then g = hifi. P is finite and R[[X]] is noetherian by 8.2.12.

8.4.2. Proposition. Let R be a principal ideal domain. Then the power series ring

R[[X]] is a unique factorization domain.

Pro**of**. Let P ⊂ R[[X]] be a minimal nonzero prime ideal. Then P + (X)/(X) =

(a) ⊂ R Suppose P = (X) and choose f = a + terms **of** positive degree ∈ P .

If g ∈ P then Xg1 = g − b1f ∈ P ∩ (X) for some b1 ∈ R. Since P is prime,

g1 ∈ P . Now Xg2 = g1 − b2f ∈ P ∩ (X) and so on. Put h = bkX k , then

g = hf. P = (f) is principal. The conditions **of** 1.5.3 are satisfied since R[[X]] is

noetherian 8.4.1.

8.4.3. Proposition. Let I ⊂ R be an ideal in a noetherian ring. Then there is a

canonical isomorphism

R[[X]]/IR[[X]] R/I[[X]]

Pro**of**. The projection R[[X]] → R[[X]]/IR[[X]] factors over R/I[[X]] since I is

finitely generated. Then there is an inverse to the homomorphism 1.9.8.

8.4.4. Corollary. If P ⊂ R is a prime ideal in a noetherian ring, then P R[[X]] ⊂

R[[X]] is a prime ideal.

8.4.5. Proposition. Let R be a noetherian ring.

(1) The inclusion R[X] ⊂ R[[X]] is a flat homomorphism.

(2) The inclusion R ⊂ R[[X]] is a faithfully flat homomorphism.

Pro**of**. (1) Let I ⊂ R[X] be an ideal and let ai ⊗ fi ∈ K = Ker(I ⊗R[X] R[[X]] → R[[X]]. Then aifi = 0. Write fi = gi + Xnhi and get ai ⊗ fi =

Xn (ai ⊗ hi). It follows that K ⊂

n Xn (I ⊗R[X] R[[X]]). I ⊗R[X] R[[X]]

is a finite R[[X]]-module and 1 + aX is a unit, so conclusion by 8.3.8. (2) The

homomorphism is flat by (1). For any maximal ideal P ⊂ R the homomorphism

RP → RP [[X]] is local and flat.

8.4.6. Exercise. (1) Show that if R[[X]] is noetherian, then R is noetherian.

98 8. NOETHERIAN MODULES

8.5. Fractions and localization

8.5.1. Proposition. Let R be a noetherian ring and U a multiplicative subset. Then

U −1 R is a noetherian ring.

Pro**of**. Any ideal is extended 4.3.6.

8.5.2. Corollary. Let R be a noetherian ring and U a multiplicative subset. If M

is a noetherian R-module, then U −1 M is a noetherian U −1 R-module.

Pro**of**. By 6.1.3 change **of** ring **of** a finite module is finite.

8.5.3. Corollary. Let R be a noetherian ring and M a finite module. Let P be a

prime ideal. Then RP is a noetherian ring and MP is a finite RP -module.

8.5.4. Proposition (Krull’s intersection theorem). Let (R, P ) be a noetherian local

ring and M a finite module. Then

P n M = 0

Pro**of**. This follows from 8.3.7.

n

8.5.5. Corollary. Let (R, P ) be a noetherian local ring and I ⊂ P an ideal. Then

I n = 0

n

8.5.6. Proposition. Let (R, P ) be a noetherian local ring and F a finite module.

The following conditions are equivalent.

(1) F is free.

(2) F is projective.

(3) F is flat.

(4) P ⊗R F → F is injective.

Pro**of**. This follows from 6.5.13.

8.5.7. Exercise. (1) Is it true that if U −1 R is noetherian, then R is noetherian?

8.6. Prime filtrations **of** modules

8.6.1. Proposition. Let R be a ring and M = 0 a nonzero module. An ideal

Ann(x) maximal in the set **of** ideals {Ann(y)|0 = y ∈ M} is a prime ideal.

Pro**of**. Let Ann(x) be a maximal annihilator. Suppose a, b ∈ R such that ab ∈

Ann(x) and b /∈ Ann(x). Then

Ann(x) ⊆ Ann(bx) = R

Consequently Ann(x) = Ann(bx) in particular a ∈ Ann(x).

8.6.2. Corollary. Let R be a noetherian ring and M = 0 a nonzero module. Then

there is x ∈ M such that Ann(x) is a prime ideal.

8.6.3. Theorem. Let R be a noetherian ring and M = 0 a finite R-module. Then

there exists a finite filtration **of** M by submodules

0 = M0 ⊂ M1 ⊂ · · · ⊂ Mr−1 ⊂ Mr = M

such that Mi/Mi−1, i = 1, . . . , r is isomorphic to an R-module **of** the form R/Pi

where Pi is a prime ideal in R.

8.6. PRIME FILTRATIONS OF MODULES 99

Pro**of**. The set **of** submodules **of** M for which the theorem is true is nonempty by

8.2.6. Let N ⊂ M be maximal in this set. Suppose N = M. By 8.2.6 applied to

M/N there is a chain N ⊂ N ′ ⊂ M such that N ′ /N is isomorphic to an R-module

**of** the form R/P ′ where P ′ is a prime ideal. This contradicts the maximality **of** N.

So N = M.

8.6.4. Corollary. Let R be a nonzero noetherian ring. Then there exists a finite

filtration **of** ideals

0 = I0 ⊂ I1 ⊂ · · · ⊂ Ir−1 ⊂ Ir = R

such that Ii/Ii−1, i = 1, . . . , r is isomorphic to an R-module **of** the form R/Pi

where Pi is a prime ideal in R.

8.6.5. Example. In Z there is a filtration

0 =⊂ (p n ) ⊂ (p n−1 ) ⊂ · · · ⊂ (p) ⊂ Z

**of** any length with factors Z/(p) for any prime number p.

8.6.6. Proposition. A ring R is artinian if and only if it is noetherian and all prime

ideals are maximal.

Pro**of**. By 7.3.13 an artinian ring has finite length and therefore noetherian. Primes

are maximal by 7.3.11. Conversely by 8.6.4 there is a finite composition series.

8.6.7. Corollary. Let R be a ring and M a module. The following are equivalent

(1) M has finite length.

(2) R/ Ann(M) is artinian and M is finite.

8.6.8. Proposition. If R ⊂ S be a finite extension. Then R is artinian if and only

if S is artinian.

Pro**of**. Suppose S is artinian. By 8.3.9 R is noetherian. By 6.6.5 any prime ideal

in R is maximal. Conclusion by 8.6.6.

8.6.9. Proposition. Let R be a noetherian ring. The number **of** minimal prime

ideals is finite.

Pro**of**. Choose a filtration 8.6.4, 0 = I0 ⊂ · · · ⊂ Ir = R with Ii/Ii−1 R/Pi,

where Pi is a prime ideal. Let P be a minimal prime ideal in R. Then (Ii/Ii−1)P

(R/Pi)P = 0 if and only if Pi ⊂ P . Thus P = Pi for some i since RP = 0.

8.6.10. Proposition. Let R be a noetherian ring such that the local rings RP are

domains for all maximal ideals P . Then R is a finite product **of** domains.

Pro**of**. Let P1, . . . , Pn be the minimal primes 8.6.9. The intersection is 0 5.4.9 and

they are comaximal since a domain has a unique minimal prime. Conclusion by

Chinese remainders 1.4.2.

8.6.11. Exercise. (1) Compute a filtration 8.6.3 **of** the Z-module Z/(36).

9

Primary decomposition

9.1. Support **of** modules

9.1.1. Definition. Let R be a ring and M a module.

(1) The set **of** prime ideals is the spectrum and denoted Spec(R).

(2) For a ring homomorphism φ : R → S restriction defines a map

(3) For a subset B ⊂ R

is a subset **of** the spectrum.

(4) The support **of** M is

φ ∗ : Spec(S) → Spec(R)

Q ↦→ φ −1 (Q)

V (B) = {P ∈ Spec(R)|MP = 0}

Supp(M) = {P ∈ Spec(R)|MP = 0}

(5) A minimal prime ideal in Supp(M) is a minimal prime **of** M.

9.1.2. Proposition. Let R be a ring.

(1) Let I ⊂ R be an ideal. Then

and is identified with Spec(R/I).

(2) Let U be a multiplicative subset.

Supp(R/I) = V (I)

Supp(U −1 R) = {P ∈ Spec(R)|P ∩ U = ∅}

and is identified with Spec(U −1 R).

Pro**of**. This is a restatement **of** 1.3.5, 5.1.5.

9.1.3. Proposition. Let 0 → N → M → L → 0 be a short exact sequence **of**

modules. Then

Supp(M) = Supp(N) ∪ Supp(L)

Pro**of**. This follows from 5.4.5.

9.1.4. Corollary. (1) Let N ⊂ M be a submodule. Then

Supp(M) = Supp(N) ∪ Supp(M/N)

(2) Given submodules N, L ⊂ M. Then

Supp(M/N ∩ L) ∪ Supp(M/N + L) = Supp(M/N) ∪ Supp(M/L)

and

Supp(M/N + L) ⊂ Supp(M/N) ∩ Supp(M/L)

Pro**of**. (2) Use the sequence 3.2.7.

101

102 9. PRIMARY DECOMPOSITION

9.1.5. Proposition. Let R be a ring and M a module.

(1) M = 0 if and only if Supp(M) = ∅.

(2) For any module

Supp(M) ⊂ V (Ann(M))

.

(3) If M is finite, then

Supp(M) = V (Ann(M))

Pro**of**. (1) See 5.4.1. (2) If MP = 0 then for u /∈ P there is x ∈ M such that

ux = 0. So Ann(M) ⊂ P . (3) Let x1, . . . , xn generate M. If Ann(M) =

∩ Ann(xi) ⊂ P then some Ann(xi) ⊂ P , so xi

1 = 0 in MP .

9.1.6. Proposition. Let N1, . . . , Nk ⊂ M be submodules such that Supp(M/Ni)∩

Supp(M/Nj) = ∅, i = j. Then the homomorphism

M/ ∩i Ni →

M/Ni

is an isomorphism.

Pro**of**. By induction on k it is enough to treat the case k = 2. By 9.1.4 the support

**of** the cokernel is empty.

9.1.7. Proposition. Let R be a ring and M, N modules.

(1)

Supp(M ⊗R N) ⊂ Supp(M) ∩ Supp(N)

(2) If M, N are finite, then

Supp(M ⊗R N) = Supp(M) ∩ Supp(N)

Pro**of**. There is an isomorphism (M ⊗R N) MP ⊗RP NP . (1) This is clear. (2)

This follows from 6.4.3.

9.1.8. Corollary. Let φ : R → S be a ring homomorphism and M an R-module.

(1) For the change **of** rings module

(2) If M is finite, then

Supp(M ⊗R S) ⊂ φ ∗−1 (Supp(M))

Supp(M ⊗R S) = φ ∗−1 (Supp(M))

9.1.9. Corollary. Let R be a ring, I an ideal in R and M a finite R-module. Then

Supp(M/IM) = Supp(M) ∩ V (I)

9.1.10. Corollary. Let R be a ring, U a multiplicative subset and M a finite Rmodule.

Then

Supp(U −1 M) = Supp(M) ∩ Spec(U −1 R)

9.1.11. Proposition. Let (R, P ) → (S, Q) be a local homomorphism and M a

finite R-module. If Supp(M) = ∅ then Supp(M ⊗R S) = ∅.

Pro**of**. The homomorphism R → S is faithfully flat 5.5.8.

9.1.12. Proposition. Let M be a finite R-module and P ∈ Supp(M). Then there

is a nonzero homomorphism M → R/P .

i

9.2. ASS OF MODULES 103

Pro**of**. The module HomR(M, R/P )P HomRP (MP , k(P )) is nonzero 7.1.4.

9.1.13. Proposition. Let R be a noetherian ring and M a finite module. M has

finite length if and only if Supp(M) consists only **of** maximal ideals.

Pro**of**. This is 8.6.7.

9.1.14. Proposition. Let R be a noetherian ring and F a finite module. If R has

only finitely many maximal ideals and FP is free **of** rank n, then F is free **of** rank

n.

Pro**of**. If R is artinian, this is clear from 7.4.5. If P1, . . . , Pk are the maximal

ideals, then choose x1, . . . , xn ∈ F giving a basis for F/P1 · · · PnF . The homomorphism

R n → F, ei ↦→ xi is an isomorphism 6.4.6 and 8.1.5.

9.1.15. Exercise. (1)

9.2. Ass **of** modules

9.2.1. Definition. Let M be an R-module. A prime ideal P ⊂ R is an associated

prime ideal **of** M if P = Ann(x) for some x ∈ M. The set **of** prime ideals

associated to M is Ass(M).

9.2.2. Proposition. Let P ⊂ R be a prime ideal.

(1) P ∈ Ass(M) if and only if there is an injective homomorphism R/P → M.

(2)

Ass(M) ⊂ Supp(M)

(3) Let 0 = N ⊂ R/P be a nonzero submodule, then

Pro**of**. This is clear from the definition.

Ass(N) = {P }

9.2.3. Proposition. Let 0 → N → M → L → 0 be a short exact sequence **of**

modules. Then

Ass(N) ⊂ Ass(M) ⊂ Ass(N) ∪ Ass(L)

Pro**of**. The left inclusion is trivial. Next let P ∈ Ass(M) such that P /∈ Ass(N).

Choose a submodule L **of** M such that L R/P . Then Ass(L ∩ N) ⊂ Ass(L) ∩

Ass(N). It follows that L ∩ N = 0, and therefore M/N contains a submodule

isomorphic to R/P .

9.2.4. Corollary. Let 0 → N → M → L → 0 be a split exact sequence **of**

modules.

Ass(M) = Ass(N) ∪ Ass(L)

9.2.5. Corollary. (1) Let N ⊂ M be a submodule. Then

(2) Let M, N be modules. Then

Ass(N) ⊂ Ass(M) ⊂ Ass(N) ∪ Ass(M/N)

Ass(M ⊕ N) = Ass(M) ∪ Ass(N)

(3) Given submodules N, L ⊂ M. Then

Ass(M/N ∩ L) ⊂ Ass(M/N) ∪ Ass(M/L) ⊂ Ass(M/N ∩ L) ∪ Ass(M/N + L)

Pro**of**. (3) Use the sequence 3.2.7.

104 9. PRIMARY DECOMPOSITION

9.2.6. Proposition. Let M be a module and P ⊂ Ass(M). Then there is a submodule

N ⊂ M such that

Ass(N) = P , Ass(M/N) = Ass(M)\P

Pro**of**. Choose by Zorn’s lemma N maximal in the set **of** submodules N ′ ⊂ M

for which Ass(N ′ ) ⊂ P. Let Q ∈ Ass(M/N) and choose N ⊂ L ⊂ M with

L/N R/Q. Then Ass(L) ⊂ P ∪ {Q}. By maximality **of** N follows that

Ass(L) ⊂ P. So Q /∈ P and Q ∈ Ass(L) ⊂ Ass(M).

9.2.7. Proposition. Let R be a noetherian ring and M a module. Then M = 0 if

and only if Ass(M) = ∅.

Pro**of**. If M = 0, then by 8.6.2 Ass(M) = ∅.

9.2.8. Corollary. Let R be a noetherian ring and f : M → N a homomorphism.

The following conditions are equivalent.

(1) f is injective.

(2) fP is injective for all prime ideals P ∈ Ass(M).

Pro**of**. Use 9.2.7 on Ker f.

9.2.9. Corollary. Let R be a noetherian ring and M a module. Then a ∈ R is a

nonzero divisor on M if and only if

a /∈ ∪ P ∈Ass(M)P

The set **of** zero divisors on M is ∪ P ∈Ass(M)P .

Pro**of**. aM is injective if and only if aMP

This happens when a /∈ ∪ P ∈Ass(M)P .

is injective for all P ∈ Ass(M), 8.2.8.

9.2.10. Proposition. Let U ⊂ R be a multiplicative subset and M a module.

(1) In general

(2) If R is noetherian

Ass(M) ∩ Spec(U −1 R) ⊂ Ass U −1 R(U −1 M)

Ass(M) ∩ Spec(U −1 R) = Ass U −1 R(U −1 M)

Pro**of**. Let P be a prime ideal. For any homomorphism R/P → M there is a

commutative diagram

R/P

M

U −1R/P

U −1M (1) If P ∈ Ass(M) and P ∩ U = ∅ then U −1 P ∈ Ass(U −1 M). (2) If U −1 P ∈

Ass(U −1 M), then P ∩ U = ∅. By 8.2.9 there is a diagram as above. R/P → M

is injective by 9.2.8. So P ∈ Ass(M).

9.2.11. Proposition. Let R be a noetherian ring and M a module. Then any minimal

prime P ∈ Supp(M) is contained in Ass(M).

Pro**of**. Assume P ∈ Supp(M) is minimal. Then the RP -module MP has support

exactly in the maximal ideal, so {P RP } = Ass(MP ). Conclusion by 9.2.10.

9.2. ASS OF MODULES 105

9.2.12. Definition. A non minimal prime ideal in Ass(M) is an embedded prime

**of** M.

9.2.13. Proposition. Let R be a noetherian ring and M a finite module. Then

Ass(M) is a finite set.

Pro**of**. Follows immediately from 9.2.4 and 8.1.5.

9.2.14. Corollary. Let R be a noetherian ring and M a finite module. The following

are equivalent

(1) M has finite length.

(2) Supp(M) consists **of** maximal ideals.

(3) Ass(M) consists **of** maximal ideals.

Pro**of**. (3) ⇒ (2): The minimal ideals in the support are maximal.

9.2.15. Lemma. Let (R, P ) be a local ring M a module. Then P ∈ Ass(M) if

and only if HomR(k(P ), M) = 0.

9.2.16. Proposition. Let R be a noetherian ring and M a finite module. For any

module N

Ass(HomR(M, N)) = Supp(M) ∩ Ass(N)

Pro**of**. By 8.2.9 HomR(M, N)P HomRP (MP , NP ). So reduce to the case

where (R,P) is local. Now

HomR(k(P ), HomR(M, N)) = HomR(M, HomR(k(P ), N))

Conclusion by Nakayama’s lemma 6.4.1 and 9.2.15.

= Hom k(P )(M ⊗R k(P ), HomR(k(P ), N))

9.2.17. Proposition. Let R be a noetherian ring and F a finite module. Assume

rank F ⊗R k(P ) = n for all primes P . Then F is locally free (projective) if and

only if FP is free for all P ∈ Ass(R).

Pro**of**. Let Q be a maximal ideal and 0 → K → R n Q → FQ → 0 exact. Ass(K) ⊂

Ass(R), so KP = 0 for all P ∈ Ass(K). By 9.2.7 K = 0.

9.2.18. Proposition. Let (R, P ) be a noetherian local ring. If there is a nonzero

finite injective module E, then R is artinian.

Pro**of**. Let Q be a prime and f : R/Q → E. If a ∈ P \Q then a R/Q is injective,

so there is f ′ : R/Q → E such that f = f ′ ◦ a R/Q. That is P HomR(R/Q, E) =

HomR(R/Q, E), so by Nakayama’s lemma 6.4.1 HomR(R/Q, E) = 0 if Q = P .

By 9.2.7 0 = HomR(R/P, E) ⊂ HomR(R/Q, E), so Q = P . R is artinian by

8.6.6.

9.2.19. Exercise. (1) Show that

(2) Show that

Ass(Z/(n)) = {(p)|p prime dividing n}

Ass(K[X, Y ]/(X) ∩ (X 2 , Y 2 )) = {(X), (X, Y )}

and point out an embedded prime.

(3) Let I ⊂ R be an ideal such that √ I = I. Show that R/I has no embedded prime

ideals.

106 9. PRIMARY DECOMPOSITION

(4) Let I, J ⊂ R be a ideals such that JRP ⊂ IRP for all P ∈ Ass(R/I). Show that

J ⊂ I.

9.3. Primary modules

9.3.1. Definition. A submodule N ⊂ M is a primary submodule or more precisely

P -primary if Ass(M/N) = {P }.

9.3.2. Proposition. A prime ideal P ⊂ R is a P -primary submodule.

Pro**of**. This is 9.2.2.

9.3.3. Proposition. Let R be a noetherian ring and M a finite module. For a

submodule N ⊂ M the following are equivalent.

(1) N ⊂ M is primary for some prime

(2) The set **of** zero divisors on M/N is contained in the radical Ann(M/N).

Pro**of**. (1) ⇒ (2): Ann(M/N) = P the set **of** zero divisors by 9.2.9. (2) ⇒ (1):

If P1, P2 ∈ Ass(M/N) then P1 ∪ P2 ⊂ Ann(M/N) ⊂ P1 ∩ P2, so P1 = P2.

9.3.4. Corollary. Let R be a noetherian ring and I ⊂ R a proper ideal. The

following are equivalent.

(1) I ⊂ R is primary for some prime.

(2) Any zero divisor in R/I is nilpotent.

9.3.5. Corollary. Let R be a noetherian ring.

(1) If an ideal I ⊂ R is P -primary then √ I = P .

(2) If the radical √ I = P is a maximal ideal, then I ⊂ R is a P -primary

submodule.

(3) A finite power P n ⊂ R **of** a maximal ideal is a P -primary submodule.

9.3.6. Proposition. Let R be a noetherian ring and M a finite module.

(1) If N ⊂ M is P -primary, then Ann(M/N) ⊂ R is P -primary.

(2) If N, N ′ ⊂ M are P -primary, then N ∩ N ′ is P -primary.

Pro**of**. (1) This follows from 9.3.3. (2) This follows from 9.2.5.

9.3.7. Proposition. Let R be a noetherian ring and M a finite module. Suppose

N ⊂ M is P -primary and U ⊂ R is multiplicative subset.

(1) If U ∩ P = ∅, then U −1 N ⊂ U −1 M is P U −1 R-primary.

(2) If U ∩ P = ∅, then U −1 N = U −1 M.

Pro**of**. This follows from 9.2.10.

9.3.8. Exercise. (1) Let K be a field. Show that (X 2 , Y ) ⊂ K[X, Y ] is (X, Y )primary.

(2) Let p be a prime number. Show that (p k ) ⊂ Z is a primary ideal.

9.4. Decomposition **of** modules

9.4.1. Definition. A submodule L ⊂ M has a primary decomposition if there

exist a family Ni ⊂ M **of** Pi-primary submodules, such that

L = N1 ∩ · · · ∩ Nn

A primary decomposition is a reduced primary decomposition if Pi = Pj for i = j

and no Ni can be excluded.

9.4. DECOMPOSITION OF MODULES 107

9.4.2. Lemma. Let R be a noetherian ring and M a finite module. For each Pi ∈

Ass(M) there is a submodule Ni ⊆ M such that Ass(Ni) = Ass(M) − {Pi} and

Ass(M/Ni)) = {Pi}. M injects

0 → M →

M/Ni

Pro**of**. The submodule Ni is given by 9.2.6. Ass(∩Ni) = ∅, so conclusion by

9.2.2.

9.4.3. Proposition. Let R be a noetherian ring and M a finite module. A proper

submodule L ⊂ M has a reduced primary decomposition

and for any such

and

is exact.

Pro**of**. Apply 9.4.2 to M/L.

L = N1 ∩ · · · ∩ Nn

Ass(M/L) = {P1, . . . , Pn}

0 → M/L →

M/Ni

9.4.4. Proposition. Let R be a noetherian ring and M a finite module. If

L = N1 ∩ · · · ∩ Nn

is a reduced primary decomposition **of** L ⊂ M and Pi is minimal in Ass(M/L),

then

and therefore uniquely determined.

Ni = M ∩ LPi

Pro**of**. Clearly Ni ⊂ M ∩ LPi . By localization Ass(M ∩ LPi /Ni) = ∅. So

equality.

9.4.5. Proposition. Let R be a noetherian ring and M a finite module. Let L ⊂ M

such that M/L = 0 has finite length. If Ass(M/L) = {P1, . . . , Pn}, then there is

a reduced primary decomposition

where

and an isomorphism

L = N1 ∩ · · · ∩ Nn

Ni = M ∩ LPi

M/L

M/Ni

Pro**of**. This follows from 9.4.3, 9.4.4 and 9.1.6.

9.4.6. Proposition. Let R be a noetherian ring and M a finite length module. If

Ass(M) = {P1, . . . , Pn}, then there is a reduced primary decomposition

where

0 = N1 ∩ · · · ∩ Nn

Ni = Pi ni M, M/Ni MPi

i

i

i

108 9. PRIMARY DECOMPOSITION

and isomorphisms

M

Pro**of**. This follows from 7.4.5.

i

MPi

M/Pi niM 9.4.7. Proposition. Let R be a noetherian ring and M a finite module. Let L ⊂ M

have a reduced primary decomposition

L = N1 ∩ · · · ∩ Nn

where Ni is Pi-primary. Assume U to be a multiplicative subset disjoint from

exactly P1, . . . Pk. Then

is a reduced primary decomposition.

U −1 L = U −1 N1 ∩ · · · ∩ U −1 Nk

Pro**of**. This follows from 9.3.7 and 9.4.3.

9.4.8. Exercise. (1) Describe the primary decomposition over a field.

9.5. Decomposition **of** ideals

9.5.1. Proposition. A proper ideal I ⊂ R has a reduced primary decomposition

and for any such

and

is exact.

Pro**of**. This is a case **of** 9.4.3.

I = Q1 ∩ · · · ∩ Qn

Ass(R/I) = {P1, . . . , Pn}

0 → R/I →

R/Qi

9.5.2. Proposition. If

I = Q1 ∩ · · · ∩ Qn

is a reduced primary decomposition and Pi is minimal in Ass(R/I), then

and therefore uniquely determined.

Pro**of**. This is a case **of** 9.4.4.

Qi = R ∩ IRPi

9.5.3. Definition. Let P be a prime ideal. The symbolic power **of** P is

P (n) = R ∩ P n RP

9.5.4. Proposition. Let

P n = Q1 ∩ · · · ∩ Qn

be a reduced primary decomposition **of** a power **of** a prime ideal P . If Ass(R/Q1) =

{P }, then

Q1 = P (n)

Pro**of**. This is a case **of** 9.4.4.

i

i

9.5. DECOMPOSITION OF IDEALS 109

9.5.5. Proposition. Let I ⊂ R be a proper ideal in a noetherian ring such that R/I

has finite length. If Ass(R/I) = {P1, . . . , Pn}, then there is a reduced primary

decomposition

I = Q1 ∩ · · · ∩ Qn

where

and an isomorphism

Pro**of**. This is a case **of** 9.4.3 and 9.5.4.

Qi = R ∩ QPi

R/I

R/Qi

9.5.6. Proposition. Let R be an artinian ring. If Ass(M) = {P1, . . . , Pn}, then

there is a reduced primary decomposition

where

and isomorphisms

Pro**of**. This is a case **of** 9.4.6.

0 = Q1 ∩ · · · ∩ Qn

Qi = Pi ni , R/Qi RPi

R

i

RPi

i

R/Pi ni

9.5.7. Proposition. Let a proper ideal I ⊂ R in a noetherian ring have a reduced

primary decomposition

I = Q1 ∩ · · · ∩ Qn

where Qi is Pi-primary. Assume U to be a multiplicative subset disjoint from

exactly P1, . . . Pk. Then

is a reduced primary decomposition.

Pro**of**. This is a case **of** 9.4.7.

U −1 I = U −1 Q1 ∩ · · · ∩ U −1 Qk

9.5.8. Example. Let R be a unique factorization domain. A factorization into

powers **of** different irreducible primes is a reduced primary decomposition **of** a

principal ideal.

9.5.9. Proposition. Let R be a noetherian ring. The following are equivalent

(1) R is reduced.

(2) RP is a field for all P ∈ Ass(R).

(3) RP is a domain for all P ∈ Ass(R).

Pro**of**. (1) ⇒ (2): The maximal ideal P RP = 0. (3) ⇒ (1): This follows from

√ 0 = ∩P ∈Ass(R)P .

9.5.10. Corollary. Let R be a reduced noetherian ring. Then all elements in

Ass(R) are minimal primes. That is, there are no embedded primes.

9.5.11. Exercise. (1) Let I ⊂ R be an ideal. Show that if P = √ I is a maximal ideal,

then I is a P -primary ideal.

(2) Let I ⊂ R be an ideal. Show that if I contains a power **of** a maximal ideal P , then I

is a P -primary ideal.

i

110 9. PRIMARY DECOMPOSITION

(3) Let K be a field and I = (X 2 , XY ) ⊂ K[X, Y ]. Show that √ I = (X), but I is not

(X)-primary.

10

Dedekind rings

10.1. Principal ideal domains

10.1.1. Lemma. Let R be a domain. The set **of** elements x ∈ M in a module with

Ann(x) = 0 is a submodule.

Pro**of**. The product **of** two nonzero elements is nonzero.

10.1.2. Definition. Let R be a domain. An element x ∈ M in a module is a torsion

element if Ann(x) = 0. By 10.1.1 the set **of** torsion elements is a submodule

T (M) ⊂ M, the torsion submodule. If T (M) = 0 then M is a torsion free

module.

10.1.3. Lemma. Let R be a domain.

(1) A flat module is torsion free.

(2) M/T (M) is torsion free.

(3) Let K be the fraction field. Then T (M) = Ker(M → M ⊗R K).

(4) If M → N → L is exact, then T (M) → T (N) → T (L) is exact.

(5) If U ⊂ R is multiplicative, then U −1 T (M) = T (U −1 M).

Pro**of**. (1) If 0 = a ∈ R and F flat, then aF is injective. (2), (4), (5) These are

clear. (3) This follows from 4.4.1.

10.1.4. Corollary. Let R be a domain and M a module. The following are equivalent.

(1) M is torsion free.

(2) MP is torsion free for all prime ideals P .

(3) MP is torsion free for all maximal ideals P .

10.1.5. Lemma. Let R be a principal ideal domain. A submodule **of** a finite free

module is free.

Pro**of**. Let F ⊂ R n be a submodule and p : R n → R the last projection. Then

p(F ) is a principal ideal and free. By induction F ∩ Ker p is free, so F F ∩

Ker p ⊕ p(F ) is free.

10.1.6. Proposition. Let R be a principal ideal domain.

(1) A torsion free module is flat.

(2) A finite torsion free module is free.

Pro**of**. (1) For a nonzero ideal (a) ⊂ R the composite R (a) → R is aM.

For a torsion free F the homomorphism aF : F (a) ⊗R F → F is injective.

So F is flat by 3.7.12. (2) Let K be the fraction field. Let F ⊂ F ⊗R K be a

torsion free submodule and suppose x1, . . . , xn ∈ F give a basis for F ⊗R K.

Then F ′ = Rx1 ⊕ · · · ⊕ Rxn ⊂ F is a free submodule such that F/F ′ is a finite

111

112 10. DEDEKIND RINGS

torsion module. There is 0 = a ∈ Ann(F/F ′ ), so F aF ⊂ F ′ . Conclusion by

10.1.5.

10.1.7. Proposition. Let R be a principal ideal domain. A finite module M decomposes

M = T (M) ⊕ F

as a direct sum **of** the torsion submodule and a finite free submodule F .

Pro**of**. The sequence 0 → T (M) → M → M/T (M) → 0 is split exact.

10.1.8. Proposition. Let R be a principal ideal domain. A finite torsion module

M has a primary decomposition

Where

M = ⊕ (p)Mp

Mp = {x ∈ M|p n x = 0, for some n}

Pro**of**. This is the primary decomposition 9.4.6.

10.1.9. Proposition. Let R be a principal ideal domain and (p) an irreducible

principal ideal. A finite torsion module M such that M = Mp has decomposition

Where n1 ≥ · · · ≥ nk.

M = R/(p n1 ) ⊕ · · · ⊕ R/(p nk )

Pro**of**. Let n1 = n be such that p n ∈ Ann(M), but p n−1 x = 0 for some x ∈ M.

The short exact sequence 0 → Rx → M → M/Rx → 0 **of** R/(p n )-modules

is split exact, since R/(p n ) Rx is an injective module 7.5.9. Conclusion by

induction on ℓR(M).

10.1.10. Exercise. (1) Show that a nonzero finite Z-submodule **of** Q is a free module

**of** rank 1.

(2) Show that a finite torsion Z-module is a finite group.

(3) Let K be a field. Show that a finite torsion K[X]-module is a finite K vector space.

(4) Let R be a noetherian domain such that every nonzero prime ideal is maximal. Let

M be a finite module. Show that the torsion submodule T (M) has finite length.

10.2. Discrete valuation rings

10.2.1. Definition. A local principal ideal domain, which is not a field, is a discrete

valuation ring. A generator **of** the maximal ideal is a local parameter or a

uniformizing parameter.

10.2.2. Proposition. Let (R, (p)) be a discrete valuation ring. Then

Pro**of**. See 8.5.5.

∩n(p n ) = 0

10.2.3. Proposition. Let (R, (p)) be a discrete valuation ring. Any nonzero ideal

is **of** the form (p n ) for a unique n = 0, 1, 2, . . . .

Pro**of**. Let 0 = x R, by 10.2.2 there is n such that x ∈ (p n ) − (p n+1 ). Since any

ideal is finitely generated it follows that a nonzero ideal is **of** the form (p n ). n is

unique by unique factorization.

10.3. DEDEKIND DOMAINS 113

10.2.4. Corollary. Let (R, (p)) be a discrete valuation ring. Any nonzero element

in the fraction field K **of** R has a unique representation up n where u is a unit and

n ∈ Z.

10.2.5. Definition. Let K be a field. A surjective map v : K\{0} → Z satisfying

(1) v(xy) = v(x) + v(x).

(2) If x + y = 0, v(x + y) ≥ min(v(x), v(y).

is a valuation on K.

10.2.6. Proposition. Let v be a valuation on a field K. Then

R = {x|v(x) ≥ 0}

is the discrete valuation ring **of** v. A local parameter is any element p, v(p) = 1.

Pro**of**. Clear from the definition.

10.2.7. Proposition. Let (R, (p)) be a discrete valuation ring with fraction field

K. The map

v : K\{0} → Z, up n ↦→ n

is a valuation and R is the discrete valuation ring **of** v.

Pro**of**. Clear from the definitions.

10.2.8. Proposition. Let R, (p) be a discrete valuation ring. A finite module M

has decomposition

Where n1 ≥ · · · ≥ nk > 0.

M = R/(p n1 ) ⊕ · · · ⊕ R/(p nk ) ⊕ R n

Pro**of**. This is a case **of** 10.1.7 and 10.1.9.

10.2.9. Exercise. (1) Let K be a field. Show that the subring K[[X 2 , X 3 ]] ⊂ K[[X]]

is not a discrete valuation ring.

10.3. Dedekind domains

10.3.1. Definition. A noetherian domain R, which is not a field, is a Dedekind

domain if all local rings RP at nonzero prime ideals are discrete valuation rings.

10.3.2. Proposition. Let R be a Dedekind domain.

(1) Any nonzero prime ideal is maximal.

(2) If U ⊂ R is multiplicative, then U −1 R is a field or a Dedekind domain.

Pro**of**. (1) (0), P are the only prime ideals in a prime ideal P . (2) This is clear

from 5.2.11.

10.3.3. Proposition. Let R be a domain which is not a field. The following conditions

are equivalent.

(1) R is a Dedekind domain.

(2) Every nonzero proper ideal in R is a product **of** finitely many maximal ideals.

Pro**of**. The primary ideals P (n) = P n . Conclusion by 9.5.4 and Chinese remainders

1.4.2.

10.3.4. Proposition. Let R be Dedekind domain.

(1) If R is a unique factorization domain then it is a principal ideal domain.

114 10. DEDEKIND RINGS

(2) If R has only finitely many maximal ideals then it is a principal ideal domain.

Pro**of**. (1) Any nonzero prime ideal is principal. Conclusion by 10.3.3. (2) Let

P, P2 . . . , Pn be the finitely many maximal ideals. Choose a ∈ P \P 2 ∪P2 · · ·∪Pn,

5.1.3. Then (a) is P primary. By 10.3.3 (a) = P k , so (a) = P . As all maximal

ideals are principal, conclusion by 10.3.3.

10.3.5. Proposition. Let R be Dedekind domain. An ideal I is generated by at

most two elements.

Pro**of**. Let Ass(R/I) = {P1, . . . , Pn} and U = R\P1 ∪ · · · ∪ Pn, then by 10.3.4

U −1 R is a principal ideal domain. By 10.1.6 U −1 R U −1 I, so choose by 8.2.9 a

homomorphism f : R → I such that U −1 f is an isomorphism. Then f is injective

9.2.8 and the ideal f(R) = (a) ⊂ R satisfies: Pi /∈ Ass(R/(a)) for any i. Let

Q1, . . . , Qm ∈ Ass(I/(a)) and choose b ∈ I\Q1 ∪ · · · ∪ Qm. b is a nonzero

divisor on I/(a) and therefore b I/(a) is an isomorphism as I/(a) has finite length.

It follows that I = (a, b).

10.3.6. Theorem. Let R be Dedekind domain.

(1) A torsion free module is flat.

(2) A finite torsion free module is projective.

(3) Any ideal is projective.

(4) Let F be a finite torsion free module. Then there is a number n and an ideal

I such that

F R n ⊕ I

Pro**of**. (1), (2), (3) These follow from 10.1.6, 6.5.13. (4) Let R have fraction field

K and assume rankK F ⊗RK = n+1. Choose a nonzero homomorphism F → R

and get by induction on n, F I1⊕· · ·⊕In+1 for nonzero ideals Ij in R. It suffices

to treat the case n = 1. Let Ass(R/I1) = {P1, . . . , Pm} and U = R\P1∪· · ·∪Pm,

then by 10.3.4 U −1 R is a principal ideal domain. By 10.1.6 U −1 I2 U −1 R, so

choose by 8.2.9 a homomorphism f : I2 → R such that U −1 f is an isomorphism.

Then f is injective 9.2.8 and the ideal f(I2) ⊂ R satisfies: Pi /∈ Ass(R/f(I2)) for

any i. It follows that I1 + f(I2) = R. Conclusion by a surjection I1 ⊕ I2 → R.

10.3.7. Proposition. Let R be a Dedekind domain. A finite module M decomposes

M = T (M) ⊕ F

as a direct sum **of** the torsion submodule and a finite torsion free submodule F .

Pro**of**. By 10.3.6 the projection M → M/T (M) splits.

10.3.8. Proposition. Let R be a Dedekind domain. A finite torsion module M has

decomposition

M = R/P n1

nk

1 ⊕ · · · ⊕ R/Pk Where P1, . . . , Pk are not necessarily distinct maximal ideals.

Pro**of**. This follows from 9.5.1 and 10.2.9.

10.3.9. Corollary. Let R be a Dedekind domain and M a finite module. Then M

has decomposition

M = R/P n1

1

⊕ · · · ⊕ R/P nk

k ⊕ Rn ⊕ Q1 · · · Ql

Where P1, . . . , Pk, Q1, . . . , Ql are not necessarily distinct maximal ideals.

10.3. DEDEKIND DOMAINS 115

10.3.10. Exercise. (1) Show that the ring Z[ √ −5] is a Dedekind domain.

(2) Show that the ring Z[ √ 5] is not a Dedekind domain.

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117

118 BIBLIOGRAPHY

A/B, 9

P -primary, 106

0-sequence, 39

abelian group, 9

addition, 9

annihilator, 27

artinian module, 87

artinian ring, 87

artinian rings, 87

ass **of** modules, 103

associated prime ideal, 103

associative, 9, 21

basis, 29

bilinear, 21

binomial formula, 10

canonical homomorphism, 59

canonical ring homomorphism, 57

Cayley-Hamilton’s theorem, 77

change **of** ring, 36

change **of** rings, 36

characteristic, 12

Chinese remainder theorem, 14

Chinese remainders, 14

coefficient, 16, 20

c**of**actor matrix, 75

cokernel, 25

colon ideal, 11, 27

comaximal ideals, 14

commutative, 9

composition series, 85

constants, 16

content **of** polynomial, 64

contracted ideal, 11

contravariant, 31

cosets, 9

Cramer’s rule, 76

decomposable, 30

decomposition **of** ideals, 108

decomposition **of** modules, 106

Dedekind domain, 113

Dedekind domains, 113

Index

119

degree, 16

derivative, 18

determinant, 75

direct product, 28

direct sum, 28, 29

discrete valuation ring, 112

discrete valuation ring **of** v, 113

discrete valuation rings, 112

distributive, 9

divisible module, 53

domain, 10

dual homomorphism, 33

dual module, 33

embedded prime, 105

essential extension, 54

evaluation, 32

evaluation map, 17

exact sequence, 39

exact sequences, 39

exactness and localization, 70

exactness **of** fractions, 60

exactness **of** hom, 48

exactness **of** tensor, 49

extended ideal, 11

factor group, 9

factor module, 24

factor ring, 11

faithfully flat, 71

faithfully flat ring homomorphism, 71

field, 10

field extension, 19

fields, 19

finite field extension, 19

finite ideal, 11

finite length, 85

finite module, 73

finite modules, 73

finite presented module, 80

finite presented modules, 80

finite ring extension, 83

finite ring homomorphism, 83

finite ring homomorphisms, 83

finite type ring, 17

120 INDEX

finite type rings, 95

finitely generated ring, 17

five lemma, 46

flat module, 54

flat modules, 54

flat ring homomorphism, 71

flat ring homomorphisms, 71

fraction field, 58

fractions and localization, 98

free module, 29

free modules, 75

Frobenius homomorphism, 12

functor, 31

Gauss’ lemma, 64

generated, 23

going-down, 72

going-up, 83

Gorenstein ring, 92

greatest common divisor, 16

Hilbert’s basis theorem, 95

homomorphism, 9, 21

homomorphism module, 30

homomorphism modules, 30

homomorphism modules **of** fractions, 63

ideal, 11

ideal generated by, 11

ideal product, 11

ideals, 11

idempotent, 15

identity, 9, 21

identity isomorphism, 9, 21

image, 25

indecomposable, 30

induced module, 37

injections, 28

injective envelope, 54

injective module, 52

injective modules, 52

irreducible element, 15

irreducible principal ideal, 15

isomorphism, 9, 21

Jacobson radical, 68

kernel, 11, 25

kernel and cokernel, 25

Krull’s intersection theorem, 96, 98

Krull’s theorem, 65

leading coefficient, 16

least common multiple, 16

length, 86

linear map, 22

local artinian ring, 91

local parameter, 112

local ring, 67

local ring homomorphism, 67

localization, 90

localization **of** modules, 68

localization **of** rings, 67

localized homomorphism, 68

localized module, 68

localized ring, 67

locally free module, 69

maximal ideal, 13

minimal prime, 101

minimal prime ideal, 66

minor, 75

module, 21

module **of** fractions, 59

modules and homomorphisms, 21

modules and submodules, 93

modules **of** fractions, 58

monic polynomial, 16

monomial, 16

multiplication, 9

multiplication **of** principal ideals, 15

multiplicative subset, 57

multiplicity, 18

Nakayama’s lemma, 78

natural homomorphism, 31

natural isomorphism, 31

negative, 9

nilpotent, 14

nilradical, 14

noetherian module, 93

noetherian ring, 94

noetherian rings, 94

noncommutative ring, 9

nontrivial idempotent, 15

nonzero divisor, 10, 22

order, 20

polynomial, 16

polynomial ring, 16

polynomials, 16

power series, 20

power series ring, 20

power series rings, 97

primary decomposition, 106

primary modules, 106

primary submodule, 106

prime fields, 19

prime filtrations **of** modules, 98

prime ideal, 13

prime ideals, 13, 65

principal ideal, 11

principal ideal domain, 15

principal ideal domains, 111

product ring, 10

projection, 9, 24

projections, 28

projective module, 50

projective modules, 50

proper ideal, 11

radical, 14

rank, 76

reduced, 14

reduced primary decomposition, 106

reflexive module, 33

residue field, 67

residue homomorphism, 68

restriction **of** scalars, 22

retraction, 41

ring, 9

ring extension, 9

ring generated, 17

ring **of** fractions, 57

rings, 9

rings **of** fractions, 57

root, 18

roots, 18

scalar multiplication, 21, 22

section, 41

short exact sequence, 40

simple module, 85

simple modules, 85

simple root, 18

snake homomorphism, 44

snake lemma, 45

spectrum, 101

split exact sequence, 42

standard basis, 29

subfield, 19

subgroup, 9

submodule, 21

submodule generated, 23

submodules and factor modules, 23

subring, 9

sum and product, 28

support, 101

support **of** modules, 101

symbolic power, 108

tensor modules **of** fractions, 62

tensor product, 33

tensor product modules, 33

tensor product ring, 38

the length, 85

the polynomial ring is factorial, 64

the snake lemma, 43

torsion element, 111

torsion free module, 111

torsion submodule, 111

total ring **of** fractions, 58

uniformizing parameter, 112

INDEX 121

unique factorization, 15

unique factorization domain, 15

unit, 10

valuation, 113

vector space, 22

windmill lemma, 46

zero, 9

zero divisor, 10, 22

zero ideal, 11

zero module, 21

zero submodule, 21