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A First Course in Noncommutative Rings

T. Y. Lam
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics 131
[Reviewed by
Fernando Q. Gouvêa
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While I was reading Lam’s First Course in Noncommutative Rings, I often found myself thinking “what a good book this is” and then wondering how to explain what made it such a good book. It’s certainly not just that the results are true and the proofs correct — that’s the minimum standard for a mathematics book. But it’s hard to put one’s finger on what it is exactly.

It helps, of course, that this is a fascinating subject. The book opens with the Wedderburn-Artin theorems about simple and semisimple rings, which is surely one of the great results of early twentieth century mathematics. Then comes the theory of the Jacobson radical, the great advance made in the middle of that century. It’s truly beautiful stuff.

And it’s also powerful, as the third chapter demonstrates. Lam deploys the theory he has developed in the first two chapters to give an elegant account of the basics of group representation theory in chapter three. He is careful to explain that he doesn’t mean to give an exhaustive account of the theory — this is, after all, a course on rings, not on representations. But the basic results are all given and illustrate how effectively the structure theory of the first two chapters can be used.

The chapters that follow continue the treatment of important ideas in the theory of noncommutative rings, including a nice introduction to the theory of division rings and a chapter on local and semilocal (noncommutative!) rings.

The book is organized as a series of “lectures,” numbered from 1 to 24, with the chapter division superimposed. Each lecture is followed by exercises. These tend to be meaty and non-trivial. All of the exercises, with solutions, appear in the companion book, Exercises in Classical Ring Theory.

But with all that, I don’t think I’ve really explained why this is such a good book. The author has excellent taste, as the outline shows. The book comes with lots of nice problems and solutions have been made available, yes.

In the end, however, it’s really about two things: mathematical insight and great sentences. Here are the opening words in chapter one:

Modern ring theory began when J. H. M. Wedderburn proved his celebrated classification theorem for finite-dimensional semisimple algebras over fields. Twenty years later, E. Noether and E. Artin introduced the Ascending Chain Condition (ACC) and the Descending Chain Condition (DCC) as substitutes for finite dimensionality, and Artin proved the analogue of Wedderburn’s Theorem for general semisimple rings. The Wedderburn-Artin theory has since become the cornerstone of noncommutative ring theory, so in this first chapter of our book it is only fitting that we devote ourselves to an exposition of this basic theory.

One is motivated to read on and armed with two important insights. First, when the chain conditions are brought in, think about the analogy to finite dimensionality. Second, the key property is going to be this “semisimple” thing, so watch for the definition to come. Elegantly done, with a bit of historical background.

The whole book is like that. Here is Lam in lecture 4, just after having defined the Jacobson radical:

In the definition of rad R above, we used the maximal left ideals of R, so rad R should be called the left radical of R, and we can similarly define the right radical of R… It turns out, fortuitously, that the left and right radicals coincide, so the distinction is, after all, unnecessary. We shall now try to prove this result: this is done by obtaining a left-right symmetric characterization of the (left) radical rad R.

Elegant and to the point. There are many ways of proving that the left-right distinction does not matter here, but of course the most elegant thing to do is precisely to prove an equivalent characterization that is left-right symmetric. Very nice.

Anyone who wants to learn about noncommutative rings should read at least the first few chapters of Lam’s book. It is the ideal introduction to the subject. 

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.

* Wedderburn-Artin Theory * Jacobson Radical Theory * Introduction to Representation Theory * Prime and Primitive Rings * Introduction to Division Rings * Local Rings, Semilocal Rings, and Idempotents * Perfect and Semiperfect Rings * References