# Introduction To Commutative Algebra

Avalon Publishing, Feb 21, 1994 - Mathematics - 138 pages
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 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

 Rings and Ideals 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 Index 128 Copyright