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

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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 proofs 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
Page 1: ELEMENTARY COMMUTATIVE ALGEBRA LECT
Page 5: Contents Prerequisites 7 1. A dicti
Page 10 and 11: 10 1. A DICTIONARY ON RINGS AND IDE
Page 22 and 23: 22 2. MODULES (5) If R is a field,
Page 24 and 25: 24 2. MODULES (1) IM = {a1y1 + ·
Page 26 and 27: 26 2. MODULES 2.3.6. Corollary. Let
Page 28 and 29: 28 2. MODULES (2) Give an example o
Page 30 and 31: 30 2. MODULES 2.4.11. Proposition.
Page 32 and 33: 32 2. MODULES Proof. By 2.5.3 the c
Page 34 and 35: 34 2. MODULES (3) The formation of
Page 36 and 37: 36 2. MODULES 2.6.12. Example. Let
Page 38 and 39: 38 2. MODULES (1) The change of rin
Page 40 and 41: 40 3. EXACT SEQUENCES OF MODULES 3.
Page 42 and 43: 42 3. EXACT SEQUENCES OF MODULES Fo
Page 44 and 45: 44 3. EXACT SEQUENCES OF MODULES an
Page 46 and 47: 46 3. EXACT SEQUENCES OF MODULES Pr
Page 54 and 55: 54 3. EXACT SEQUENCES OF MODULES Pr
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56 3. EXACT SEQUENCES OF MODULES Pr
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58 4. FRACTION CONSTRUCTIONS 4.1.4.
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60 4. FRACTION CONSTRUCTIONS 4.2.7.
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62 4. FRACTION CONSTRUCTIONS and ge
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64 4. FRACTION CONSTRUCTIONS 4.6. T
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66 5. LOCALIZATION (2) For a prime
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68 5. LOCALIZATION (2) Any local ri
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70 5. LOCALIZATION 5.4. Exactness a
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72 5. LOCALIZATION (1) φ is flat.
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74 6. FINITE MODULES 6.1.7. Corolla
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76 6. FINITE MODULES (4) Let A be a
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78 6. FINITE MODULES Let the n × n
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80 6. FINITE MODULES 6.5. Finite Pr
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82 6. FINITE MODULES 6.5.9. Corolla
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84 6. FINITE MODULES (1) E is an in
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86 7. MODULES OF FINITE LENGTH 7.2.
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88 7. MODULES OF FINITE LENGTH Proo
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90 7. MODULES OF FINITE LENGTH (5)
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92 7. MODULES OF FINITE LENGTH 7.5.
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94 8. NOETHERIAN MODULES Proof. Use
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96 8. NOETHERIAN MODULES 8.3.2. Cor
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98 8. NOETHERIAN MODULES 8.5. Fract
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9 Primary decomposition 9.1. Suppor
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9.2. ASS OF MODULES 103 Proof. The
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9.2. ASS OF MODULES 105 9.2.12. Def
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9.4. DECOMPOSITION OF MODULES 107 9
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9.5. DECOMPOSITION OF IDEALS 109 9.
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10 Dedekind rings 10.1. Principal i
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10.3. DEDEKIND DOMAINS 113 10.2.4.
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10.3. DEDEKIND DOMAINS 115 10.3.10.
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118 BIBLIOGRAPHY
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120 INDEX finite type rings, 95 fin
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Commutative algebra - Department of Mathematical Sciences - old ...

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