Reference request: introduction to commutative algebra

Question

My goal is to pick up some commutative algebra, ultimately in order to be able to understand algebraic geometry texts like Hartshorne's. Three popular texts are Atiyah-Macdonald, Matsumura (Commutative Ring Theory), and Eisenbud. There are also other books by Reid, Kemper, Sharp, etc. Can someone outline the differences between these texts, their relative strengths, and their intended audiences?

I am not listing my own background and strengths, on purpose, (a) so that the answers may be helpful to others, and (b) I might be wrong about myself, and I want to hear more general opinions than what might suite my narrow profile (e.g. If I said "I only like short books", then I might preclude useful answers about Eisenbud, etc.).

Answer 1

For a down-to-earth introduction, see A Singular Introduction to Commutative Algebra (also here). See also Reference book for commutative algebra from MO and a list but without comments.

Answer 2

Eisenbud's book ("Commutative Algebra with a view towards algebraic geometry") is really gentle, and has good explanations together with lots of good illustrations. It is however huge, so you shouldn't really try to read the whole book.

Atiyah-MacDonald is no doubt the best mathematics book I've read, but it was a pain working through it because it is so densely written.

Edit: Some of the reasons I like Atiyah-MacDonald: * Tons of exercises - and really good ones too. * Short, concise language (this makes reading proofs a bit harder, but it pays off later when you're using the book as reference). * Very good choice of topics covered.

Answer 3

I would recommend:

(1) Firstly, one should study field theory and Galois theory fairly thoroughly. The main reasons are:

a. Fields are the best understood examples of commutative rings from an ideal-theoretic point of view (a field has exactly two ideals) and field theory often motivates many important concepts in commutative algebra, e.g., modules (analogue in field theory: vector spaces) and integral extensions (analogue in field theory: algebraic extensions); also polynomial rings over fields are the best understood types of polynomial rings and are one of the main objects of study in algebraic geometry.

b. The applications of commutative algebra to algebraic number theory, for example, is very much based on Galois theory.

(2) Once one has a solid understanding of field theory and Galois theory, one can start learning commutative algebra. There are many good books on commutative algebra at the basic level. I recommend Atiyah and Macdonald's "An Introduction to Commutative Algebra" for the following reasons:

a. The book presents commutative algebra in a very elegant manner. I can assure you that if you read the entire book (~ 130 pages) and do all the exercises, you will have a very solid knowledge of commutative algebra.

b. The exercises are excellent and introduce the reader to many important concepts in commutative algebra not treated in the text, e.g. the spectrum of a ring, affine schemes, faithful flatness, direct limits, Hilbert's Nullstellensatz, Noether's normalization lemma etc. It is highly recommended that one does, or at least looks at, all of the exercises since approximately half of the material in the book is treated in the exercises. Many exercises have hints (which are almost always complete solutions) and thus the book is suitable for self-study.

Unfortunately, I have not read too many other introductory books on commutative algebra. "Algebra: A Graduate Course" by Martin Isaacs is also a good introduction to commutative algebra; however, the book is not one on commutative algebra purely. Similarly, Serge Lang's "Algebra" is also a good introduction.

(3) I think that there are three main choices for commutative algebra reading after Atiyah and Macdonald: "Commutative Algebra" by Hideyuki Matsumura, "Commutative Ring Theory" by Hideyuki Matsumura, and "Commutative Algebra: With a View Toward Algebraic Geometry" by David Eisenbud. Chapter IV of EGA is also a good reference if you are comfortable with the idea of reading French. However, I have not read any of these books (although I will do so soon) and thus I cannot comment further. Note that Matsumura's "Commutative Algebra" has very few exercises.

I hope this helps!

Answer 4

It's a bit late. But since no one has mentioned it, I would mention Gathmann's lecture notes on Commutative Algebra (https://www.mathematik.uni-kl.de/~gathmann/class/commalg-2013/commalg-2013.pdf).

The exposition is excellent. The content is comparable to Atiyah-McDonald, but contains much more explanation. It emphasizes the geometric intuitions throughout the lectures. For example, the chapters on integral ring extension and Noetherian normalization have one of the best expositions of the geometric pictures behind these important algebraic concepts that I have read among several introductory books on commutative algebra. Chapters usually begin with a very good motivation and give many examples.