The first eight chapters of this slender (150 pages) book offer a well-written and reader-friendly introduction to some of the more introductory parts of noncommutative ring theory, specifically modules (Noetherian and Artinian, simple and semisimple, projective and injective), the Jacobson radical, the Weddeburn-Artin structure theorem, the Hopkins-Levitzki theorem, and Maschke’s theorem from group representation theory (phrased in terms of the semisimplicity of the group ring). Prerequisites for the book include good previous courses in abstract and linear algebra. The language of category theory is introduced, and continually used throughout the text; the author feels that studying a relatively concrete example (the category of modules) helps the student adapt to the more general theory.

This material is, as noted above, fairly introductory, and the author’s approach is not rushed; Lam’s book *A First Course in Noncommutative Rings*, for example, discusses the Weddeburn-Artin theorem in the very first chapter, whereas this book doesn’t get to that result until chapter 7.

The level of sophistication picks up in the last two chapters of the book. Chapter 9 discusses exchange modules and exchange rings, which are generalizations of semisimple modules and rings. Chapter 10 looks at a small part of the more general (and still unsolved) question of when a group ring K[G] is semiprimitive. The highlight of this chapter is a proof of a theorem of Rickart, namely that if K is the field of complex numbers, K[G] is semiprimitive for all groups G. This material is very interesting, in that it illustrates some nice connections between analysis and algebra. (One might not expect C* algebras to be alluded to in a book on noncommutative ring theory.)

Although quite well done, this book may suffer from something of an audience problem. The author describes the book as being suitable for undergraduates, but it should be kept in mind that he teaches in the United Kingdom, where “suitable for undergraduates” has a different meaning than it does here in the United States. Given my own experiences teaching much less sophisticated abstract algebra to undergraduates, I cannot help but wonder if many are really prepared to plunge into a fairly concise book that covers things like the Weddeburn-Artin theory. And even those that might make a go of it will likely not get the chance: at most non-elite universities on this side of the pond, there are very few if any courses that include modules, let alone category theory, as part of the syllabus.

What about using this text for a graduate course? Here we have a potential reverse problem: it is possible that instructors of a graduate course might prefer a less elementary account of the subject, with perhaps wider coverage. Lam’s aforementioned book, for example, spends an entire 50-page chapter on group representation theory, as an application of some of the ideas developed previously in the text. Lam also discusses a number of other topics not covered here, including the Jacobson density theorem and Kothe’s conjecture. Moreover, the style of writing in Lam’s book seems more appropriate for a graduate course. At one point in Mathieu’s book, for example, he wants to prove the result that in a ring with identity, if \( 1 – ab \) is invertible then so is \( 1 – ba \). He does this by writing down a formula for the inverse of \( 1 – ba \) (in terms of the inverse of \( 1 – ab\) ) and then showing that it is an inverse by doing the (trivial) multiplication in detail. He does not explain how one might discover this formula for the inverse. Lam, by contrast, leaves the result as an exercise in First Course and then, in his companion book *Exercises in Classical Ring Theory* gives a delightful explanation (learned, he tells us, from Kaplansky) of how one might find this formula and then leaves the verification that the formula works as an exercise to the reader, calling it a “breeze”.

However, I don’t want to end on a negative note. This is, as noted, a well-written book, one that might find good use as a text for an elementary graduate course or a rather sophisticated undergraduate one. Faculty members may also find this to be a useful book to have on their shelves. I’m glad it’s on mine.

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