Commutative ring theory / Hideyuki Matsumura ; translated by M. Reid

From the preface: "In publishing this English edition I have tried to make a rather extensive revision. Most of the mistakes and insufficiencies in the original edition have, I hope, been corrected, and some theorems have been improved. Some topics have been added in the form of appendices to individual sections. Only Appendices A, B and C are from the original. The final section, §33, of the original edition was entitled `Kunz' theorems' and did not substantially differ from a section in the second edition of my previous book Commutative algebra [second edition, Benjamin/Cummings, Reading, MA, 1980; MR0575344 (82i:13003)], so I have replaced it by the present §33. The bibliography at the end of the book has been considerably enlarged, although it is obviously impossible to do justice to all of the ever-increasing literature.'' (MathSciNet)

Bibliogr. p. 298-316. Index

This text book is meant for graduate students and covers all the topics which are considered standard today. The presentation is very lucid and a pleasure to read through. Concessions are made to Algebraic Geometry in the selection of topics and theorems. For instance there are many results about fibre dimensions and several on smoothness and the treatment compares well with the author’s earlier book "Commutative Algebra" (New York 1970; Zbl 0211.06501; second ed. 1980). There are a few appendices which are certainly welcome diversions. The bibliography is vast, but many of the papers are only alluded to. But I have no doubt that a serious student will benefit immensely from this. The indexing is fairly complete - a notable and somewhat baffling omission is that of Noether’s normalisation lemma, though the lemma itself is proved in the text. Sections end with several exercises; hints and solutions are provided at the end of the book. The book is almost entirely error-free. May be the most confusing expression appears on p. 82, "a product of zero ideals" and a careful reader will not find it misleading. This is definitely a worthwhile book to read, especially if you are a student. (Zentralblatt)