The intent of this book is to provide a rather quick introduction to the theory of commutative algebra. We quote from the authors' introduction: "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 on commutative algebra such as those of 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 [D. G. Northcott, Ideal theory, Cambridge Univ. Press, London, 1953; MR 15, 390] and our treatment is substantially different in that, following the modern trend, we put more emphasis on modules and localization."

The general style of the book is concise and to the point. In the authors' words, "The lecture-note origin of this book accounts for the rather terse style, with little general padding, and for the condensed account of many proofs. We have resisted the temptation to expand it in the hope that the brevity of our presentation will make clearer the mathematical structure of what is by now an elegant and attractive theory. Our philosophy has been to build up to the main theorems in a succession of simple steps and to omit routine verifications."

The authors have used the methods of homological algebra but do not pursue the subject to any great depth. We again quote from the introduction, "Anyone writing now on commutative algebra faces a dilemma in connection with homological algebra, which plays such an important part in modern developments. A proper treatment of homological algebra is impossible within the confines of a small book: on the other hand, it is hardly sensible to ignore it completely. The compromise we have adopted is to use elementary homological methods---exact sequences, diagrams, etc.---but to stop short of any results requiring a deep study of homology."

Chapter 1 covers rings, ideals, radials, extensions, and contractions. In Chapter 2 various properties of modules are determined. Rings and modules of fractions are considered in Chapter 3. The classical primary decomposition theorems are proved in Chapter 4. Chapter 5 is devoted to integral dependence and valuations. The going-up theorem and going-down theorem are given in this chapter.

The next chapter deals with chain conditions in modules. Chapters 7 and 8 cover Noetherian rings and Artin rings, respectively. Discrete valuation rings and Dedekind domains are examined in Chapter 9. Chapter 10 deals with topologies, completions, filtrations, and graded rings. Hilbert functions and dimension theory are developed in Chapter 11.

A substantial number of exercises have been provided at the end of each chapter. Some of them are simple and others are rather difficult. Some hints and some complete solutions are provided for the more difficult problems.

Reviewed by J. A. Johnson