This post is a list of various books in commutative algebra (mostly called 'Commutative Algebra') with some comments. It is mainly geared towards students who might want to read about the subject on their own, though others might find it useful. It is not meant to be a comprehensive list, as it reflects the random process of my coming into contact with them.
- David Eisenbud, Commutative Algebra: with a View Toward Algebraic Geometry. This text is a popular, comprehensive near 800 page tome. It contains the basics of the subject along with thorough treatments of various topics such as dimension theory and homological methods. Once I tried to read it many years ago in graduate school but I found that the material is so comprehensive that it was difficult to choose a short path to the topics in which I was interested. However, it is an outstanding reference.
- Irving Kaplansky, Commutative Rings. In my mind, this is the ultimate introduction to commutative algebra. It is not comprehensive but in its 169 pages of text, brings the reader to understand zero divisors on modules, regular rings, and homological methods quickly and easily. It is short enough to read in its entirety in a few months, and contains many exercises at the right level. This book is probably best for someone who knows a little about commutative and homological algebra and wants to see some interesting things quickly.
- Michael F. Atiyah and Ian G. MacDonald, Introduction to Commutative Algebra. This is a true classic. It is an extremely rapid, bare-bones 127 page boot camp for the basics commutative algebra. Most popular amongst the "read minimally, do mostly exercises" crowd. Clearly written, but I found the large number of gothic letters and the chapter on primary decomposition confusing.
- Gert-Martin Greuel and Gerhard Pfister, A Singular Introduction to Commutative Algebra. This book covers basics, free resolutions, normalization, primary decomposition, and Hilbert functions. Moreover, it does this by illustrating examples with the free and open-source software Singular, and thus many of the examples are concrete. Those who want to focus on the algorithmic side of commutative algebra should enjoy this book.
- Bourbaki, Commutative Algebra. A text that is mostly of historical interest, and is not easy to read. Yet it should be remembered because it contains facts that are sometimes difficult to find elsewhere.
- N.S. Gopalakrishnan, Commutative Algebra. A little-known text of just under 300 pages that covers localisation, Noetherian rings, integral extensions, Dedekind domains, completions, homology, Krull dimension, and regular local rings. It is more idiosyncratic in its coverage, which makes it more interesting than the generic standard syllabus, and could be a very nice read for self-study.
- Huishi Li, An Introduction to Commutative Algebra: From the Viewpoint of Normalization. This short 188-page book presents the topic of normalization and how it plays a role in algebraic geometry, though it requires few prerequisites. This book would be ideal for someone who is interested in learning algebraic geometry and commutative algebra and yet has no experience with it, because it presents a particular slice of these topics while taking care of all the prerequisites.