## Introduction To Commutative Algebra (Google eBook)This book grew out of a course of lectures given to third year undergraduates at Oxford University and it has the modest aim of producing a rapid introduction to the subject. It is designed to be read |

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### Contents

2 | |

3 | |

4 | |

NILRAOICAL AND JACOBSON RADICAL | 6 |

OPERATIONS ON IDEALS | 7 |

EXTENSION AND CONTRACTION | 10 |

EXERCISES | 11 |

Modules | 18 |

THE GOINGUP THEOREM | 62 |

INTEGRALLY CLOSED INTEGRAL DOMAINS THE GOINGDOWN THEOREM | 63 |

VALUATION RINGS | 66 |

EXERCISES | 68 |

Chain Conditions | 75 |

EXERCISES | 79 |

Noetherian Rings | 81 |

PRIMARY DECOMPOSITION IN NOETHERIAN RINGS | 83 |

SUBMODULES AND QUOTIENT MODULES | 19 |

OPERATIONS ON SUBMODULES | 20 |

DIRECT SUM AND PRODUCT | 21 |

FINITELY GENERATED MODULES | 22 |

EXACT SEQUENCES | 23 |

TENSOR PRODUCT OF MODULES | 25 |

RESTRICTION AND EXTENSION OF SCALARS | 28 |

EXACTNESS PROPERTIES OF THE TENSOR PRODUCT | 29 |

ALGEBRAS | 30 |

TENSOR PRODUCT OF ALGEBRAS | 31 |

EXERCISES | 32 |

Rings and Modules of Fractions | 37 |

LOCAL PROPERTIES | 41 |

EXTENDED AND CONTRACTED IDEALS IN RINGS OF FRACTIONS | 42 |

EXERCISES | 44 |

Primary Decomposition | 51 |

EXERCISES | 56 |

Integral Dependence and Valuations | 60 |

EXERCISES | 85 |

Artin Rings | 90 |

EXERCISES | 92 |

Discrete Valuation Rings and Dedekind Domains | 94 |

DISCRETE VALUATION RINGS | 95 |

DEDEKIND DOMAINS | 96 |

TOPOLOGIES AND COMPLETIONS | 102 |

FILTRATIONS | 106 |

GRADED RINGS AND MODULES | 107 |

THE ASSOCIATED GRADED RING | 112 |

EXERCISES | 114 |

11 Dimension Theory | 117 |

DIMENSION THEORY OF NOETHERIAN LOCAL RINGS | 120 |

REGULAR LOCAL RINGS | 124 |

TRANSCENDENTAL DIMENSION | 125 |

EXERCISES | 126 |

Index | 128 |

### Common terms and phrases

a-filtration a-topology abelian group Artin ring Artinian Chapter closure coefficients commutative algebra composition series coprime Corollary Dedekind domain Deduce defined denote dimension direct sum discrete valuation ring equation exact sequence example exists field of fractions finitely-generated A-module following are equivalent fractional ideal graded ring Hausdorff hence induction injective integral domain integrally closed intersection inverse irreducible isomorphism Jacobson radical kernel lemma mapping maximal element maximal ideal minimal primary decomposition minimal prime ideal module homomorphism morphism multiplicatively closed subset nilpotent nilradical Noetherian local ring Noetherian ring non-zero element p-primary polynomial ring primary decomposition primary ideals prime ideals belonging principal ideal Proof Proposition Prove quotient residue field resp ring and let ring homomorphism S~lA satisfies Show Spec subgroup submodule subring subspace Suppose surjective tensor product theorem topology unique zero ideal zero-divisor