Herstein’s book is a guided tour through a gallery of masterpieces. The author’s style is always elegant and his proofs always enlightening.

From the preface: “This book is not intended as a treatise on ring theory. Instead, the intent is to present a certain cross-section of ideas, techniques and results”. In fact, many portions of the theory are not included (e.g. Lie algebras, Azumaya algebras, representation theory of algebras) and the choice of the topics depends mainly on the author’s taste.

The book is based on a series of lectures given at Bowdoin College in the summer of 1965, addressed to a group of mathematicians teaching at various colleges and smaller universities. The prerequisites are a good familiarity with basic ring theory: homomorphisms, quotients, ideals.

The first two chapters introduce, with remarkable limpidity, the basic structure theory: the Jacobson radical, the semisimple Artinian rings, the primitive rings and the Density Theorem. The six remaining chapters, essentially independent from each other, touch upon several aspects of the highest importance. They are short surveys (some 20 pages), each entirely self-contained and with a list of references for further reading. As is perfectly logical and pertinent in a book on non-commutative algebras, chapter 3 offers a collection of commutativity theorems, including some intriguing generalizations of Wedderburn’s theorem (i.e.any finite division ring is commutative).

But this small book is does not merely give a list of big results; the focus is in fact on the methods and the ideas from which the theorems flow. On page 54, on the basis of the structure theory given in the previous pages, Herstein sketches a general method to attack any given problem in (non-commutative) ring theory: 1) prove the theorem for division rings; 2) pass to primitive rings (dense subrings of rings of linear transformations); 3) induct the result into matrix rings; 4) tie together to get the result for semisimple rings; 5) climb back through the Jacobson radical to conclude. As he shows in the third chapter, this procedure is especially efficacious in proving that appropriately conditioned rings are commutative.

I had a lot of pleasure when I first read this book, while I was an undergraduate student attending to a course given by C. Procesi at the University of Rome. Today, I appreciate even more the author’s mastery and real gift for exposition. Like J.-P. Serre and J. Milnor, I. N. Herstein belonged to that category of remarkable mathematicians with an unfailing sense of beauty.

Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. He may be reached at mainardi2002@yahoo.fr.