Module Theory From now on, R will denote an arbitrary ring with 1 ...

<strong>Module</strong> <strong>Theory</strong>
<strong>From</strong> <strong>now</strong> on, R will denote an arbitrary ring with 1.
Definition. A (left) module M over R is an abelian group together with an action of R on
M (a mapping R × M → M) such that for all a, b ∈ R, andallx, y ∈ M, (a+b)x=ax + bx,
a(x + y) =ax + ay, (ab)x = a(bx) and1x=x.Aright module is defined similarly. If R is
commutative, there is no need for a distinction between left and right.
Note that 0R · x =0M and r · 0M =0M.
Examples: R is an R-module. R × R is an R-module with r · (s, t) =(rs, rt). {0} is an
R-module for any ring R. A vector space over a field is a module over that field. Any abelian
group is a Z-module. In fact, we shall prove the standard theorems for abelian groups in
terms of modules over a PID. If f : R → S is a ring homomorphism, then S is an R-module
via r · s = f(r)s. Inparticular,everyringcanbeviewedasaZ-module.
Definition. Let M be an R-module. A submodule N of M is an additive subgroup of M
such that RN ⊆ N. For a left ideal I ⊳ R, define IM = { aimi | ai ∈ I, mi ∈ M }. Itis
a submodule of M. The submodules of R are its left ideals.
Given an R-module M and submodule N, we define a module structure on the group
M/N as follows: for r ∈ R, m + N ∈ M/N, set r(m + N) =rm + N. Check that this is
well-defined and the module axioms hold. Call M/N the quotient or factor module.
Definition. Let M,M ′ be R-modules. An R-module homomorphism f : M → M ′ is a
homomorphism of abelian groups which is R-linear; i.e. f(rm) =rf(m).
Examples:
1M : M → M, the identity mapping, 0: M → M ′ , the zero mapping and the canonical
M → M/N for a submodule N

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4. Submodules of M/N correspond to submodules of M containing N.
Theorem. In the category of R-modules, direct products and direct sums (coproducts) exist.
For any subset S of an R-module M, the set N = { n
1 risi | ri ∈ R, si ∈ S } is
the submodule of N generated by S. M is finitely generated if it has a finite number of
generators. For m ∈ M, Rm is a principal submodule of M.
Proposition. Let M be an R-module, N1,N2 submodules of M. M ∼ = N1 ⊕ N2 iff M =
N1 + N2 and N1 ∩ N2 = {0}.
Theorem. Let I be any set. The free R-module on the set I is the coproduct R (I) .
Definition. Let M be an R-module. A subset S ⊆ M is a basis for M if every element of
M can be written uniquely as a linear combination of elements of S.
Remarks. The free modules are the modules that have bases. Direct sums of free modules
are free since R (I) ⊕ R (J) ∼ = R I ˙∪J) . The zero module (0) is the free module on the empty
set.
Example. The free abelian groups are the groups of the form Z (I)
. If |I| = n,wehave
n
i=1 Z is the free abelian group on n generators. The Z-module Z/mZ is thus an abelian
group which is not free.
Proposition. Every module can be written as the quotient of a free module.
Characterization of finitely generated modules over a PID.
Theorem. Let R be a PID, F a finitely generated free module over R and M a submodule
of F . Then M is free with the number of generators less than or equal to the number of
generators of F . (This is true without the hypothesis of finitely generated [L, Theorem
III.7.1].)
Invariant Basis Property. Any two bases of F have the same number of elements.
The theorem fails for principal ideal rings (not domains): take R = Z/6Z and M =
{0, 2, 4}. The theorem fails for UFD’s: take R = R[x, y] andM=theideal(x, y).

Proposition. Let R be a commutative ring and F a finitely generated free module. Given a
basis {e1,...,en}, and elements fi = n j=1 aijej, theset{f1,...,fn} is a basis iff det(aij)
is an invertible element of R.
Lemma. Let R be a PID, a1,...an ∈R, d=gcd(a1,...an).Then(a1,...an)=(d).
Theorem. Let R be a PID and F = Re1 ⊕···⊕Ren be free. Let f = n
i=1 aiei ∈ F .Then
fis part of a basis iff gcd(a1,...an)=1.
Corollary. f is part of a basis iff there exists a module homomorphism φ: F → R such
that φ(f) =1.
Stacked Basis Theorem. Let F be a finitely generated free module over a PID R and
M

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p1,...pk and positive integers e1,...,ek and an integer n ≥ 0 such that G is isomorphic to
⊕Zn .
Z p e 1
1 ⊕···⊕Z p e k
k
If n =0,thenGis finite. If k =0,thenGis free. In particular, every finitely generated
abelian group is a direct sum of cyclic groups.
Noetherian, Artinian modules and the Jordan-Hölder Theorem.
For this section, R is any ring with 1.
Theorem. Let M be a left R-module. The following are equivalent:
(1) All submodules of M are finitely generated.
(2) (ACC) Given any sequence of submodules N1 ⊆ N2 ⊆···, it must eventually become
constant.
(3) Any nonempty collection of submodules has a maximal element.
<strong>Module</strong>s satisfying the conditions of the previous theorem are called Noetherian. If all
left ideals of R are finitely generated, then R is a left Noetherian ring.
A module is called Artinian if the following hold
(1) (DCC) Every descending chain of submodules eventually becomes constant.
(2) Every collection of submodules has a minimal element.
These two conditions are equivalent by an analogous proof to that in the Noetherian case.
Theorem. Let N

Definition. A composition series for an R-module M is a finite sequence of submodules
0=N0 ⊆N1 ⊆···⊆Ns =M such that each quotient Ni/Ni−1 is simple. Call s the length
of the composition series and Ni/Ni−1 the composition factors. We say M has finite length
if it has a composition series.
Theorem. M has finite length iff M satisfies both ACC and DCC.
Our proof of the Jordan-Hölder Theorem for groups carries over (but is easier as we don’t
have to worry about normality) to this setting.
Jordan-Hölder Theorem. Let M be an R-module with a composition series of length s.
Then all composition series for M have length s and the composition factors are the same
(up to permutation and isomorphism).
References
[J2] N. Jacobson, Basic Algebra II, W. H. Freeman and Co., 1980.
[L] S. Lang, Algebra, 3rd ed., Addison-Wesley, 1993.
[S] M. Steinberger, Algebra, PWS Publishing Co., Boston, 1994.
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