In his introduction, the author points out that “In the last one hundred years there has been a vast literature in noncommutative theory, and our goal here has been to find as much of a common framework as possible. Much of the theory can be cast in terms of representations into matrix algebras”.

This book is a companion volume to Graduate Algebra: Commutative View*,* which I reviewed some time ago. In this impressive tome of 600 pages, Rowen gives a fairly complete overview of many of the major themes in noncommutative algebra. As is clear from the quoted citation, the emphasis is put on the representation theory of finite-dimensional algebras.

The first six chapters are devoted to the structure of rings, introducing fundamental notions such as simple and semisimple modules, the Jacobson radical, primitive rings, and tensor products. Thirty pages of exercises conclude this first part of the book, which may already serve as a textbook for an introductory course. The chapter on free groups and graphs looks a bit disconnected from the rest (but, I must say, it is very well written and is my favourite one!).

In the second part, the structure theorems are applied to the representation theory of finite-dimensional algebras, the focus being specifically on group algebras of finite groups and the finite-dimensional complex Lie algebras. Of course, the representation theory of Lie algebras deserves a whole course in itself, and the chapter contained in this volume is to be intended only as an initiation. The same remark applies to the chapters of the third, and last, part of the book, where more advanced topics are touched upon: PI-algebras, the Brauer group, homological algebra and Hopf algebras. These are vast theories, occupying an important place in current research, and Rowen makes a valuable effort to develop them from scratch. However, I think that the chapter on Hopf algebras would have profited from a short section on affine group schemes, in the style of Waterhouse’s book; in fact, functors and algebraic groups are introduced elsewhere in the book, and affine groups schemes provide a very concrete illustration of the representation theory of Hopf algebras.

The book is largely self-contained (so you don’t need to buy the first volume).

A pleasant feature of this volume is that it contains a gentle exposition of some results that are usually limited to technical publications (and often hard to read for the neophyte). For instance: Zelmanov’s solution to the Burnside problem, or Tits’s alternative.

Many different paths can be chosen to make up a course based on this book; the first six chapters should necessarily form the core of the course, followed by additional topics chosen among the remaining chapters.

To conclude, as I already said about the first volume, I believe that this book will be a valuable textbook and a reliable reference for graduate students.

Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are mainly Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. At present, he works in a "classe préparatoire" in Geneva. He may be reached at mainardi2002@gmail.com.