## Introduction To Commutative AlgebraThis 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 by students who have had a first elementary course in general algebra. On the other hand, it is not intended as a substitute for the more voluminous tracts such as Zariski-Samuel or Bourbaki. We have concentrated on certain central topics, and large areas, such as field theory, are not touched. In content we cover rather more ground than Northcott and our treatment is substantially different in that, following the modern trend, we put more emphasis on modules and localization. |

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### Contents

2 | |

IDEALS QUOTIENT RINGS | 3 |

PRIME IDEALS AND MAXIMAL IDEALS | 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 |

128 | |

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### Common terms and phrases

A-algebra a-filtration A-module homomorphism a-topology abelian group Artin ring Artinian chain of prime Chapter closure coefficients commutative algebra composition series coprime Corollary Dedekind domain Deduce defined denote dimension direct limit equation exact sequence example exists field of fractions finitely generated A-module finitely-generated A-module following are equivalent fractional ideal graded ring Hausdorff hence homomorphism f induces injective integral domain integrally closed intersection inverse irreducible isomorphism Jacobson radical kernel lemma Let f Let x1 maximal element maximal ideal minimal primary decomposition minimal prime ideal modules morphism multiplicatively closed subset nilpotent nilradical Noetherian local ring Noetherian ring p-primary polynomial ring power series primary decomposition primary ideals prime ideals belonging principal ideal Proof Proposition Prove quotient residue field resp ring and let ring homomorphism satisfies Show Spec Spec(A subgroup submodule subring subspace Suppose surjective tensor product theorem topology unique zero ideal zero-divisor