This first volume of Rowen's textbook is aimed mainly towards the commutative-algebraic framework for algebraic geometry (the second volume, Graduate algebra: Non-Commutative view, will be motivated by representations of groups). It consists of three major parts. The first one (Modules) describes a structure of finitely generated modules and composition series. The second one (Affine algebras and Noetherian rings) includes Galois theory and the Krull dimension theory. In the third part (Applications to geometry and number theory), the reader finds an introduction to algebraic geometry and other theoretical applications of commutative algebra. The book is consistently organized in several layers (main text, supplements, appendices), which makes it a valuable source for readers of various levels, from graduate students to researchers. Though the textbook is (in theory) self-contained, the reader has to have a good background in undergraduate algebra. As a result of the author’s lifelong teaching experience and several years of maturation of the text, this is a valuable and readable textbook on modern commutative algebra.

Reviewer:

dst