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

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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
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ELEMENTARY COMMUTATIVE ALGEBRA LECT
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Contents Prerequisites 7 1. A dicti
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Prerequisites The basic notions fro
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10 1. A DICTIONARY ON RINGS AND IDE
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22 2. MODULES (5) If R is a field,
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24 2. MODULES (1) IM = {a1y1 + ·
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26 2. MODULES 2.3.6. Corollary. Let
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28 2. MODULES (2) Give an example o
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30 2. MODULES 2.4.11. Proposition.
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32 2. MODULES Proof. By 2.5.3 the c
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34 2. MODULES (3) The formation of
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36 2. MODULES 2.6.12. Example. Let
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38 2. MODULES (1) The change of rin
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40 3. EXACT SEQUENCES OF MODULES 3.
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42 3. EXACT SEQUENCES OF MODULES Fo
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44 3. EXACT SEQUENCES OF MODULES an
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46 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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Page 72 and 73: 72 5. LOCALIZATION (1) φ is flat.
Page 74 and 75: 74 6. FINITE MODULES 6.1.7. Corolla
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Page 78 and 79: 78 6. FINITE MODULES Let the n × n
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Page 82 and 83: 82 6. FINITE MODULES 6.5.9. Corolla
Page 84 and 85: 84 6. FINITE MODULES (1) E is an in
Page 86 and 87: 86 7. MODULES OF FINITE LENGTH 7.2.
Page 88 and 89: 88 7. MODULES OF FINITE LENGTH Proo
Page 90 and 91: 90 7. MODULES OF FINITE LENGTH (5)
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Page 94 and 95: 94 8. NOETHERIAN MODULES Proof. Use
Page 96 and 97: 96 8. NOETHERIAN MODULES 8.3.2. Cor
Page 98 and 99: 98 8. NOETHERIAN MODULES 8.5. Fract
Page 101 and 102: 9 Primary decomposition 9.1. Suppor
Page 103 and 104: 9.2. ASS OF MODULES 103 Proof. The
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Page 111 and 112: 10 Dedekind rings 10.1. Principal i
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Commutative algebra - Department of Mathematical Sciences - old ...

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