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**Commutative ring theory. Transl. from the Japanese by M. Reid.**
*(English)*
Zbl 0603.13001

Cambridge Studies in Advanced Mathematics, 8. Cambridge etc.: Cambridge University Press. XIII, 320 p. Ł 30.00; $ 49.50 (1986).

[The Japanese original appeared at Kyoritsu Shuppan Co. Ltd. (Tokyo 1980).]

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.

The author says in his introduction that this book was originally supposed to have been written by Masao Narita, whose life was tragically cut short. The author has included results on UFD’s, Picard groups etc. from lectures of M. Narita as a tribute to him. The excellent translation from the original Japanese is by Miles Reid.

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.

The author says in his introduction that this book was originally supposed to have been written by Masao Narita, whose life was tragically cut short. The author has included results on UFD’s, Picard groups etc. from lectures of M. Narita as a tribute to him. The excellent translation from the original Japanese is by Miles Reid.

Reviewer: N.Mohan-Kumar

### MSC:

13-02 | Research exposition (monographs, survey articles) pertaining to commutative algebra |

13-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra |

13Axx | General commutative ring theory |

14A05 | Relevant commutative algebra |