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Functional identities. (English) Zbl 1132.16001

Frontiers in Mathematics. Basel: Birkhäuser (ISBN 978-3-7643-7795-3/pbk). xii, 272 p. (2007).
The monograph under review is the first book that treats exclusively the topic of functional identities. Functional identities, in their present form, appeared some 20 years ago. Since then the theory has been in constant and thorough development, it resulted in solving several classical problems due to Herstein and concerning Lie maps in associative rings.
The book is written by leading specialists in the area of functional identities and is based mostly on their own research plus a large number of topics treated by K. Beidar and by Mikhalev. The book is intended for ring theorists, however specialists in nonassociative algebras, operator theory, linear algebra, mathematical physics may (and will) find it interesting and useful. The book is reasonably self-contained and requires from the reader only knowledge of the basics of Ring theory. The initial parts of the book as well as the appendices contain almost all of the further pre-requisites.
The book is divided into three parts. The first one contains the basic definitions and results about functional identities. All this information is provided based on a lot of examples that try to make the readers’ life somewhat easier. Further on the authors give an idea of the concept of \(d\)-freeness in rings. This seems quite a good shot since the formal treatment of \(d\)-free sets is given in the second part of the book.
In fact Part 2 is the centre of the monograph; after the formal introduction of the \(d\)-freeness it deals with quasi-polynomials, and further constructs \(d\)-free sets in (semi-)prime rings. It should be mentioned that this second part is rather hard to read. Here the reader could observe that the first part would be very helpful in going through the main results of Part 2.
The last Part 3 is devoted to applications of the theory developed so far. It starts with the solutions of Herstein’s conjectures concerning Lie maps and goes on with related results about Jordan maps in associative rings. Next the authors discuss linear preserver problems: like commutativity, normality, and so on. Finally they consider some further applications to classes of nonassociative algebras like Lie admissible algebras, and Poisson algebras.
The book is well written. It can serve as an excellent introduction to the topic of functional identities (especially its first part), and as an exhaustive source of reference (its second part). The specialists in the area would find its third part also very useful.

MSC:

16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras
16R50 Other kinds of identities (generalized polynomial, rational, involution)
16N60 Prime and semiprime associative rings
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16W20 Automorphisms and endomorphisms
16W25 Derivations, actions of Lie algebras
15A04 Linear transformations, semilinear transformations
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
17C50 Jordan structures associated with other structures
47B47 Commutators, derivations, elementary operators, etc.
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