There are two books by Matsumura on commutative algebra. The earlier one is called Commutative Algebra and is frequently cited in Hartshorne. The more recent version is called Commutative Ring Theory and is still in print. In the preface to the latter, Matsumura comments that he has replaced a section from a previous (Japanese?) edition because it "did not substantially differ from a section in the second edition of my previous book Commutative Algebra." This suggests that Matsumura considered the two books complementary, and certainly did not intend Commutative Ring Theory to replace Commutative Algebra.

Basically, my question is

Why are there two books, by the same author, apparently on the same subject?

A more practical question is whether both books are equally appropriate as references when reading a book like Hartshorne. However, I am curious to know the answer to the broader question as well.

  • $\begingroup$ Someone told me that all of the material of one of them is incorporated into the other one. $\endgroup$ Commented May 20, 2010 at 19:57
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    $\begingroup$ If you read the preface of Comm. Ring Theory you will learn why he wrote the second book. He was replacing another author who, well, read the preface. $\endgroup$
    – KConrad
    Commented May 20, 2010 at 20:01
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    $\begingroup$ The first book has a marvelous development of excellence (chapter 13); the 2nd says almost nothing about it. Neither entirely subsumes the other, but the 2nd covers "more" stuff. The 2nd is entirely sufficient for Hartshorne. Maybe the 1st is also (?). $\endgroup$
    – BCnrd
    Commented May 20, 2010 at 20:27
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    $\begingroup$ KConrad: At the end of the introduction, Matsumura says that he was replacing a friend who had a tragic early death due to illness. However, it would have been absurd for Matsumura to attempt this unless the planned book was substantially different from the previous book. Incidentally, I was at first confused in reading your comment since this statement is in the introduction, not the preface, which is about why Matsumura felt it necessary to produce a second edition of the same book. $\endgroup$ Commented May 20, 2010 at 20:48
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    $\begingroup$ As a note, I think Hartshorne's Algebraic geometry references Commutative algebra because Commutative ring theory (1986) was written after the release of Algebraic geometry (1977) $\endgroup$ Commented Aug 14, 2019 at 8:13

1 Answer 1


By comparing the tables of contents, the two books seem to contain almost the same material, with similar organization, with perhaps the omission of the chapter on excellent rings from the first, but the second book is considerably more user friendly for learners. There are about the same number of pages but almost twice as many words per page. The first book was almost like a set of class lecture notes from Professor Matsumura's 1967 course at Brandeis. Compared to the second book, the first had few exercises, relatively few references, and a short index. Chapters often began with definitions instead of a summary of results. Numerous definitions and basic ring theoretic concepts were taken for granted that are defined and discussed in the second. E.g. the fact that a power series ring over a noetherian ring is also noetherian is stated in the first book and proved in the second. The freeness of any projective modules over a local ring is stated in book one, proved in the finite case, and proved in general in book two. Derived functors such as Ext and Tor are assumed in the first book, while there is an appendix reviewing them in the second. Possibly the second book benefited from the input of the translator Miles Reid, at least Matsumura says so, and the difference in ease of reading between the two books is noticeable. Some arguments in the second are changed and adapted from the well written book by Atiyah and Macdonald. More than one of Matsumura's former students from his course at Brandeis which gave rise to the first book, including me, themselves prefer the second one. Thus, while experts may prefer book one, for many people who are reading Hartshorne, and are also learning commutative algebra, I would suggest the second book may be preferable.

Edit: Note there are also two editions of the earlier book Commutative algebra, and apparently only the second edition (according to its preface) includes the appendix with Matsumura's theory of excellent rings.


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