Basic Books on Rings and Modules

General Theory of Rings and Modules

Lambeck, Rings and Modules

This is a very nice, small, readable book. Most of all, it is interesting. It probably represents the strongest influence on the graduate algebra course I teach.

P. M. Cohn, Algebra

3 volumes, covering undergraduate algebra, standard graduate topics, and advanced topics. Horrendously expensive.

I. M. Isaacs, Algebra, a Graduate Course

This covers many of the important topics in both commutative and non-commutative ring theory in quite a bit of detail. I've only skimmed through it, but it looks like a good alternative to Hungerford or Lang.

Classical Theory of Artinian Rings

Jacobson, The Theory of Rings (AMS Mathematical Surveys)

This is very readable and has a lot of good stuff that's too old fashioned to be included in many more modern books.

Herstein, NonCommutative Rings [Carus Monographs]

This is somewhat comparable to the Jacobson book above. It's not quite as interesting, in my opinion, but it has a good exposition of the classical theory.

Curtis and Reiner, Representations of Groups and Associative Algebras

The original edition of this book is very nice, because it has good information on a wide variety of topics, such as Dedekind domains, modules over artinian rings, and the like. Later on, though, it gets pretty specialized.

Jans, Rings and Homology (Chapter I)

A very interesting presentation of the basic Wedderburn theory. The rest of the book is also quite interesting, but a little too specialized for a beginning algebra student.

Van der Waerden, Algebra, vol II (Fifth Edition)

This book has a lot of really good material in it, especially about the classic theory of non-commutative artinian rings (Chapters 13 and 14). It's really essential to get the seventh edition, though.
Note also the concise presentation of the Riemann-Roch Theorem in Chapter 19.

Bourbaki, Algebra, Chapter VIII

It goes without saying that Bourbaki has their own take on the theory. Some really good classical stuff can be found here more readily than in most other sources.

Lang, Algebra

Everyone finds Lang hard to read because he's so concise, but his material is very well organized and his proofs are generally quite good, once you manage to understand them. I'm listing him here primarily for Chapters 17 and 18.

Commutative Rings

Zariski-Samuels, Commutative Algebra

This is the book I first learned algebra from. It's readable and it really makes the subject interesting. I wish that there were a book like this for the non-commutative theory. (I think that Jacobson's AMS notes, mentioned above, probably come the closest.)

Kaplansky, Commutative Rings

A very small book, fairly readable. It covers the basics and a number of more specialized results. It's especially good for those who are interested in non-noetherian rings. Unfortunately, it's organized rather poorly, which makes it hard to use as a reference book. There's a list of theorems in the back, with all the page numbers given. Otherwise, you'd never be able to find anything in the book.

Bourbaki, Commutative Algebra

It takes a lot of dedication to work one's way through all seven chapters of this, but I recommend doing it. As always with Bourbaki, the beginning seems really easy, but after a while the going gets impossibly difficult because of the constant need to thumb back to the earlier results they keep citing. However if you're willing to do this, you will seldom find yourself actually hung up, since every little detail is filled in.

Matsumura, Commutative Algebra

A fairly condensed treatment, and I doubt that anyone would call it interesting. But it has a lot of essential material in it. It's more of a reference than a book that one reads from cover to cover.
A bit advanced for beginners.

Nagata, Local Rings (Chapter 1)

Also pretty hard reading, but it's got a lot of important results.