Subject: Commutative Algebra

Recommendation: *Introduction to Commutative Algebra* by Atiyah & MacDonald

Contenders: the introductory chapters of *Commutative Algebra With a View Towards Algebraic Geometry* by Eisenbud and the commutative algebra chapters of *Algebra* by Lang.

Atiyah & MacDonald is a short book that covers the essentials of Commutative Algebra, while most books cover significantly more material. So this review should be seen as comparing Atiyah & MacDonald to the corresponding chapters of other Commutative Algebra books. There are a few reasons why *Introduction to Commutative Algebra* is better than most other books:

Better abstractions. The abstractions Atiyah & MacDonald use (especially towards rings and ideals) are simply more broadly applicable and make several proofs simpler. Conversely other books tend to use an older set of abstractions which make the same proofs significantly more complex.

Exercise-driven approach. Atiyah & MacDonald's exercises are beautifully structured so that you build up important parts of the theory yourself. There's a very satisfying feeling of castle-buildng: each exercise draws upon your understanding of the previous problem, and they come together to form very nice results. Many books can give you the feeling of understanding Commutative Algebra, but this one helps you discover it, which is much more enjoyable and provides a much deeper understanding.

The right kind of conciseness. Atiyah & MacDonald's book is short because they cover a limited range of topics, but they do cover all the essential tools that are widely used. In contrast most books tend to bloat by trying to cover too many things, or tend to leave out critical parts of the theory.

*2 points [-]