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.
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