For a one-semester introduction to commutative algebra, this new textbook offers some reader-friendly organization and exposition choices. It starts with some very basic topics concerning commutative rings and modules, usually treated in an undergraduate abstract algebra course, including some standard results from field theory up to the Galois correspondence and finite abelian extensions. The expositional narrative falls more on the wordy side, appropriate for self-study and with a good balance and choice on what to leave as exercises making the core narrative self-contained.

The chosen topics are classical and emphasize the close interrelations with algebraic geometry and algebraic number theory, with a nod to the computational side. Some novel topics include a chapter on class numbers of rings of algebraic integers using the embeddings of these rings as lattices in RnRn , allowing the author to use geometric techniques to derive the required bounds. There is also a chapter explicitly spelling out the lexicon that translates the commutative algebra concepts and results into the algebro-geometric realm.

With judicious choices, the book can be used as a standard introduction to commutative algebra with the usual orientation towards algebraic geometry or as an introduction to algebraic number theory.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx