This book is an introductory text to commutative algebra with the idea also of being a guide to the algorithmic branch of the subject. In particular some part of the text is devoted to computational methods. The book is divided in four parts. The first part is devoted to a geometric interpretation of some basic concepts like Hilbert's Nullstellensatz, Notherian and Artinian rings and modules and the Zariski topology. The notion of Krull dimension is discussed in part II. In particular heights of an ideal, localization, integral ring extensions and normal rings, Noether nornalization theorem and Krull's principal ideal theorem are the main topics in this part . The third part is devoted to computational methods and Gröbner bases, and the Buchberger algorithm for computing Gröbner bases is presented in detail with applications to compute elimination ideals, an algorithm to compute the image of a morphism of affine varieties. The notion of Hilbert function and Hilbert series and their relation with dimension Is also algorithmically presented. The last part of the text is devoted to (regular) local rings and local properties of varieties being the Dedekind domains case studied in detailed. As a result of various courses teaching experience, this is a valuable and readable textbook on modern commutative algebra. It contains a huge number of exercises and it appeals to geometric intuition whenever possible. It can be highly recommended for independent reading or as material for preparation of courses.

Reviewer:

Alejandro Melle Hernandez