This recent book is a welcome addition to the textbook literature on the subject of (mostly noncommutative) ring theory, and should be looked at carefully by anybody who has occasion to teach a course on the subject and wants a text that is less demanding of the reader than, say, Lam’s A First Course in Noncommutative Rings.
The book divides naturally into two halves of roughly equal length. I will refer to them as the “basic course” and the “topics course”. The former consists of the first six chapters of the book and culminates, in chapter six, with discussions of the Jacobson Density theorem, Weddeburn-Artin theory, the Jacobson radical and prime radical, and Hopkins’ theorem. The five chapters leading up to this flurry of results in classical ring theory take the reader through a review of the basic definitions concerning rings and modules, direct products and direct sums, free modules (and results concerning invariance of basis number), an introduction to category theory, Hom and the tensor product as functors, Noetherian and Artinian rings and modules, free modules over a PID, and injective, projective and flat modules.
Most or all of this material in these six chapters is found, in one form or another, in many of the standard graduate algebra textbooks such as those by Dummit and Foote, Hungerford, Rotman, or Lang, but I suspect that in many cases the instructor of a year-long introductory graduate algebra course does not get to cover all of this material in any kind of depth.
The author of this text is clearly interested in making this material, and what follows, as accessible as possible to a graduate student. The prerequisites for this book seem to be limited to a basic understanding of introductory abstract algebra and set theory (at the level of Zorn’s Lemma; cardinal and ordinal numbers are discussed in an Appendix). Contrast this, for example, with Lam’s book, which, though very well written, assumes more background on the part of the reader and which begins, in the first chapter, with Weddeburn-Artin theory. The “basic course” part of the text (perhaps supplemented by material from the second half of the book if the students are well prepared) could serve as the text for a semester-long course in ring theory.
Two things that are not covered in this first half of the book should perhaps be noted. First, although the structure theorem for free modules over a PID is proved, there is no discussion of how, for example, this theory can be applied to such things as canonical forms for matrices. It seems a pity that this is not mentioned at all, even as an extended exercise. Second, a number of books on ring theory (for example, the book by Lam cited above, as well as Beachy’s Introductory Lectures on Rings and Modules and Herstein’s Carus Monograph Noncommutative Rings) include a chapter on group representation theory — a fairly natural segue, given that group representation theory involves the study of an associated ring (the group ring over the field) which is, in many cases, semi-simple (by Maschke’s theorem). This book omits any reference to group representations.
The “topics course” part of the book consists of the six remaining chapters, which seem to be fairly independent of one another (though there is no “dependence chart” for the various chapters). The first four of these chapters discuss such topics as injective envelopes and projective covers, rings and modules of quotients and Goldie’s theorem, graded rings and modules, and reflexive modules. The final two discuss homological algebra — the first introducing the basic constructs (projective and injective resolutions, Ext and Tor, etc.) and the second illustrating the use of homological methods in ring theory (via various concepts of dimension).
This book has all the attributes of an excellent text. The writing is clear and reader-friendly, and there are both good examples and a reasonable number of exercises of varying difficulty. A lengthy bibliography lists 46 books and 26 journal articles (not all in English), so an interested reader who has finished this book should be well-positioned to pursue other topics in ring theory. It is highly recommended, both as a text for ring theory courses at the various levels described above, or as an accessible resource for graduate students with an interest in pursuing research in ring theory.
Mark Hunacek (firstname.lastname@example.org) teaches mathematics, and a course in engineering law, at Iowa State University.