In 1950 Wilhelm Specht asked whether every T-ideal of a free F-algebra on the set of letters X is finitely based. An affirmative answer implies that every identity of a PI-algebra is a consequence of a finite set of identities. Various aspects of the Specht problem have been studied by many researchers. The main purpose of this monograph is to present a complete and concise answer to the Specht question.

The first six chapters of the book develop tools needed for the Kemer proof of the Specht conjecture in characteristic zero. The choice and exposition of the topics of this part of the book (such as the Schirschov height theorem, the Razmyslov-Kemer-Braun theorem for the nilpotency of the Jacobson radical of finitely generated PI-algebras, the development of Kemer polynomials, and the theory of Grassmann algebras and their connection to superalgebras) is guided by this aim. Chapter 7 contains counterexamples to the Specht question in positive characteristics and related theory. The final five chapters of the monograph are devoted to further development of themes that were established on the way to the Kemer proof. This part of the book deals with notions of Hilbert series and Gelfand-Kirillov dimension as well as the theory of cocharacters, trace identities, and the general theory of identities. The book ends with exercises, the list of main theorems, examples and counterexamples, and a list of open problems. All topics of the monograph are well-arranged and developed in a clear way. The book is suitable not only as a useful reference for researchers but also as part of a course on PI-algebras for graduate students.

Reviewer:

jzem