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Do you know any book or an online source that contains exercises on ring theory? I've solved some exercises of Lang's Algebra and Dummit & Foote's Abstract Algebra but there is a huge gap between these two. I need a graduate level problem book but not as hard as Lang. Thanks in advance.

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This will be limited to commutative rings, but some great resources are the Commutative Algebra and Exercises chapters of (the Stacks Project) – Dylan Wilson Jan 2 '12 at 22:44
Also, Atiyah-Macdonald... and Matsumura. But these are all commutative stuff. – Dylan Wilson Jan 2 '12 at 22:44
Jacobson's Basic Algebra II has plenty of material. – lhf Jan 2 '12 at 23:41
up vote 7 down vote accepted

How about Exercises in Classical Ring Theory and Exercises in Basic Ring Theory?

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Also by T.Y. Lam:… – user641 Jan 2 '12 at 23:22
If you have access to SpringerLink, you can download Exercises in Classical Ring Theory. – Anononym Jan 7 '12 at 1:34

Harry C. Hutchins, Examples of Commutative Rings has many interesting examples (and/or counterexamples) which may serve as good exercises, with hints and literature references. The book was based upon Hutchins' 1978 Chicago thesis (under Kaplansky). It was apparently intended as a companion to Kap's classic textbook Commutative Rings (many references refer to Kap's book). There is also a 3 page list of errata, updates,... dated July 1983, which is distributed with the book. Below is the AMS Math Review.

Hutchins, Harry C. 83a:13001 13-02
Examples of commutative rings.
Polygonal Publ. House, Washington, N. J., 1981. vii+167 pp. ISBN 0-936428-05-8

The book is divided into two parts: a brief sketch of commutative ring theory which includes pertinent definitions along with main results without proof (but with ample references), and Part II, the 180 examples. The examples do cover a very large range of topics. Although most of them appear elsewhere, they are enhanced by a fairly complete listing of their properties. Example 67, for instance, is M. Hochster's counterexample to the polynomial cancellation problem, and it lists a number of properties of the two rings that were not given in the original paper Proc. Amer. Math. Soc. 34 (1972), no. 1, 81 - 82; MR 45 #3394. Some of the examples appear more than once, since many rings exhibit more than one interesting property. ($\rm\:R = K[x, y, z]\:$ is used in Examples 6 and 22.) The examples are grouped into areas, but a drawback is that these have not been labeled and separated off. In addition, the Index is for Part I and definitions only, and this means that searching for a specific example with certain properties can be time consuming.

The book can be used as a supplement to one of the standard texts in commutative ring theory, and it does appear to complement the monograph by I. Kaplansky Commutative rings, Allyn and Bacon, Boston, Mass., 1970; MR 40 #7234; second edition, Univ. Chicago Press, Chicago, Ill., 1974; MR 49 #10674.

Reviewed by Jon L. Johnson

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In the UK most (leading) unis provide problem sets openly.

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While probably true, I don't see how this answer is very helpful to the OP. Are you suggesting that s/he move to the UK and enroll in a (leading) university there? – Pete L. Clark Jan 3 '12 at 3:26
... a slightly less difficult task would be utilizing google's searching facilities to see that for instance, you can get example sheets (a.k.a problem sets) at practically all 'leading' unis. For some reason, the unis that - while all equivalent - are looked upon as being 'average' (whatever that means) do not provide this access, instead requiring, for isntance, a username and password. – Adam Jan 3 '12 at 4:23

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