Commutative Ring Theory (Cambridge Studies in Advanced Mathematics) Review

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Commutative Ring Theory (Cambridge Studies in Advanced Mathematics) ReviewMatsumura has achieved a great success with this book. The first nine chapters contain all of the algebra used in Hartshorne's _Algebraic Geometry_, and a reader who masters all of that material will be well on the way to a solid understanding of algebraic geometry. At the same time, the writing is clear and concise, and the exercises (which the reader should of course do!) are enlightening. In this regard, one should compare Matsumura's book with Eisenbud's _Commutative Algebra with a View Toward Algebraic Geometry_, which has a similar scope but takes up 800 pages. Both books are beautifully written, but while Matsumura prefers a more subtle way of teaching the reader the tricks of the trade, Eisenbud makes explicit commentary on his methods and intuition. Another helpful comparison between the two is in their fundamental approach: Matsumura tends to use universal constructions which are not computationally effective, while Eisenbud seems to keep the computer in mind. Thus, whereas Eisenbud studies the module of differentials using generators and relations, Matsumura uses the universal properties of representable functors.
Unfortunately, this book is not sufficient for a first introduction to commutative algebra, as the pace can be brisk for a reader without the mathematical maturity to appreciate Matsumura's condensed style. (One could read this book after reading Atiyah and MacDonald's _Introduction to Commutative Algebra_.) Nevertheless, for a reader seeking a solid understanding of the commutative algebra necessary for (modern) algebraic geometry, there is no better reference than Matsumura's book.
A minor gripe: the formatting is visually slightly confusing -- the announcements of Theorems, Corollaries, Lemmas, Remarks, Proofs, etc. are all set in italic typeface, so there is not much to orient the eye on the page.Commutative Ring Theory (Cambridge Studies in Advanced Mathematics) Overview

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