Commutative algebra - Department of Mathematical Sciences - old ...

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Elementary <strong>Commutative</strong> Algebra H.A. Nielsen
Page 1: ELEMENTARY COMMUTATIVE ALGEBRA LECT
Page 6 and 7: 6 CONTENTS 5.3. Localization of mod
Page 9 and 10: 1 A dictionary on rings and ideals
Page 11 and 12: 1.2. IDEALS 11 1.2. Ideals 1.2.1. D
Page 13 and 14: 1.3. PRIME IDEALS 13 since a prime
Page 15 and 16: 1.5. UNIQUE FACTORIZATION 15 1.4.3.
Page 17 and 18: 1.6. POLYNOMIALS 17 1.6.4. Proposit
Page 19 and 20: 1.8. FIELDS 19 1.8. Fields 1.8.1. D
Page 21 and 22: 2 Modules 2.1. Modules and homomorp
Page 23 and 24: 2.2. SUBMODULES AND FACTOR MODULES
Page 25 and 26: 2.3. KERNEL AND COKERNEL 25 2.3. Ke
Page 27 and 28: 2.3. KERNEL AND COKERNEL 27 Proof.
Page 29 and 30: 2.4. SUM AND PRODUCT 29 2.4.4. Defi
Page 31 and 32: There is induced a homomorphism of
Page 33 and 34: 2.6. TENSOR PRODUCT MODULES 33 2.5.
Page 35 and 36: 2.6. TENSOR PRODUCT MODULES 35 2.6.
Page 37 and 38: 2.7.4. Proposition. The constructio
Page 39 and 40: 3 Exact sequences of modules 3.1. E
Page 41 and 42: 3.1. EXACT SEQUENCES 41 3.1.8.
Page 43 and 44: 3.2. THE SNAKE LEMMA 43 3.1.16. Exa
Page 45 and 46: 3.2. THE SNAKE LEMMA 45 3.2.4. Theo
Page 47 and 48: Proof. Look at the two diagrams 0 0
Page 49 and 50: 3.3.5. Proposition. Given a sequenc
Page 51 and 52: 3.5. PROJECTIVE MODULES 51 projecti
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Proof. Let M → N be injective. Th
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3.7. FLAT MODULES 55 3.7.4. Proposi
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4 Fraction constructions 4.1. Rings
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4.2. MODULES OF FRACTIONS 59 4.2.2.
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4.3. EXACTNESS OF FRACTIONS 61 Proo
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4.5. HOMOMORPHISM MODULES OF FRACTI
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5 Localization 5.1. Prime ideals 5.
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5.2. LOCALIZATION OF RINGS 67 5.2.
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5.3. LOCALIZATION OF MODULES 69 5.3
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(3) FP is flat for all maximal idea
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6 Finite modules 6.1. Finite Module
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6.2. FREE MODULES 75 (2) Let K be a
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6.3. CAYLEY-HAMILTON’S THEOREM 77
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(2) P MP = MP for all prime ideals
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Proof. By 3.1.13 there is a split e
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6.6. FINITE RING HOMOMORPHISMS 83 a
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7 Modules of finite length 7.1. Sim
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7.3. ARTINIAN RINGS 87 7.2.9. Propo
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7.3. ARTINIAN RINGS 89 Proof. (1) L
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7.5. LOCAL ARTINIAN RING 91 (1) M
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8 Noetherian modules 8.1. Modules a
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8.3. FINITE TYPE RINGS 95 8.2.9. Pr
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8.4. POWER SERIES RINGS 97 8.3.11.
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8.6. PRIME FILTRATIONS OF MODULES 9
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102 9. PRIMARY DECOMPOSITION 9.1.5.
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104 9. PRIMARY DECOMPOSITION 9.2.6.
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106 9. PRIMARY DECOMPOSITION (4) Le
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108 9. PRIMARY DECOMPOSITION and is
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110 9. PRIMARY DECOMPOSITION (3) Le
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112 10. DEDEKIND RINGS torsion modu
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114 10. DEDEKIND RINGS (2) If R has
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Bibliography A. Altman and S. Kleim
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A/B, 9 P -primary, 106 0-sequence,
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projections, 28 projective module,
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