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.

Table of Contents

Chapter 1 Rings and Ideals

Chapter 2 Modules

Chapter 3 Rings and Modules of Fraction

Chapter 4 Primary Decomposition

Chapter 5 Integral Dependence and Valuations

Chapter 6 Chain Conditions

Chapter 7 Noetherian Rings

Chapter 8 Artin Rings

Chapter 9 Discrete Valuation Rings and Dedekind Domains

Chapter 10 Completions

Chapter 11 Dimension Theory