*(English)*Zbl 0943.00001

This is the second edition of the formidable book under the same title (see Prentice-Hall 1991; Zbl 0751.00001). The first four parts cover Group theory up to Sylow theorem, Semidirect products, Nilpotency and solvability; Ring theory up to PID’s and UFD’s, and polynomial rings; Modules and Vector spaces up to projective, injective and flat modules, tensor, symmetric and exterior algebras, modules over PID’s and canonical forms of matrices; Field theory and Galois theory.

Part 5 considers in more detail commutative and homological algebra and algebraic geometry. The last Part 6 is an introduction to the Representation theory of finite groups.

The book is more or less self contained and it can serve as a textbook for undergraduate and graduate courses in Algebra. On the other hand, due to the encyclopaedical nature of the book (it is some 898 pages) it is impossible to cover all its contents even in a year course. But the lecturer could choose several parts and/or chapters for a course. It must be noted especially the abundance of (carefully selected) exercises in the book.

The main differences between the first and the second edition are the following. First, the authors have added two new chapters (15 and 16) that treat the basics of Commutative algebra: Affine varieties, Hilbert Nullstellensatz, Localization, Prime spectrum, and Discrete valuation rings and Dedekind domains. The material about representations of groups has moved to a separate part. There are several added sections – the authors discuss tensor products of modules and exact sequences, topics that are continued in the part concerning Homological algebra.

This second edition of the book is much better organized than the first. It is complete and could serve as a base for various courses in Algebra and as a reference by graduate students. The only “weak” point of the book is its volume that could indeed frighten undergraduates. But on the other hand the book covers almost any topic usually taught as a basic course in Algebra. And therefore it is not possible to make this shorter.