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.
Author(s): Michael Atiyah, Ian G. Macdonald
Publisher: Westview Press
Year: 1994
Language: English
Commentary: Covers, 2 level bookmarks, OCR, paginated.
Pages: 140
Introduction
Notation and Terminology
1 Rings and Ideals
RINGS AND RING HOMOMORPHISMS
IDEALS. QUOTIENT RINGS
ZERO-DIVISORS. NILPOTENT ELEMENTS. UNITS
PRIME IDEALS AND MAXIMAL IDEALS
NILRADICAL AND JACOBSON RADICAL
OPERATIONS ON IDEALS
EXTENSION AND CONTRACTION
EXERCISES
2 Modules
MODULES AND MODULE HOMOMORPHISMS
SUBMODULES AND QUOTIENT MODULES
OPERATIONS ON SUBMODULES
DIRECT SUM AND PRODUCT
FINITELY GENERATED MODULES
EXACT SEQUENCES
TENSOR PRODUCT OF MODULES
RESTRICTION AND EXTENSION OF SCALARS
EXACTNESS PROPERTIES OF THE TENSOR PRODUCT
ALGEBRAS
TENSOR PRODUCT OF ALGEBRAS
EXERCISES
3 Rings and Modules of Fractions
LOCAL PROPERTIES
EXTENDED AND CONTRACTED IDEALS IN RINGS OF FRACTIONS
EXERCISES
4 Primary Decomposition
EXERCISES
5 Integral Dependence and Valuations
INTEGRAL DEPENDENCE
THE GOING-UP THEOREM
INTEGRALLY CLOSED INTEGRAL DOMAINS. THE GOING-DOWN THEOREM
VALUATION RINGS
EXERCISES
6 Chain Conditions
EXERCISES
7 Noetherian Rings
PRIMARY DECOMPOSITION IN NOETHERIAN RINGS
EXERCISES
8 Artin Rings
EXERCISES
9 Discrete Valuation Rings and Dedekind Domains
DISCRETE VALUATION RINGS
DEDEKIND DOMAINS
FRACTIONAL IDEALS
EXERCISES
10 Completions
TOPOLOGIES AND COMPLETIONS
FILTRATIONS
GRADED RINGS AND MODULES
THE ASSOCIATED GRADED RING
EXERCISES
11 Dimension Theory
HILBERT FUNCTIONS
DIMENSION THEORY OF NOETHERIAN LOCAL RINGS
REGULAR LOCAL RINGS
TRANSCENDENTAL DIMENSION
EXERCISES
Index
Back Cover
