Functional Identities

by Bresar, Matej; Chebotar, Mikhail A.; Martindale, Wallace S., III

• ISBN13: 9783764377953
• ISBN10: 376437795X
• Format: Paperback
• Copyright: 2007-07-04
• Publisher: Birkhauser

Summary

The theory of functional identities (FIs) is a relatively new one - the first results were published at the beginning of the 1990s, and this is the first book on this subject. An FI can be informally described as an identical relation involving arbitrary elements in an associative ring together with arbitrary (unknown) functions. The goal of the general FI theory is to describe these functions, or, when this is not possible, to describe the structure of the ring admitting the FI in question. This abstract theory has turned out to be a powerful tool for solving a variety of problems in ring theory, Lie algebras, Jordan algebras, linear algebra, and operator theory. The book is divided into three parts. Part I is an introductory one. Part II is the core of the book. It gives a full account of the general FI theory, which is based on the concept of a d-free set; various constructions and concrete examples of d-free sets are given, and FI's on d-free sets are thoroughly studied. Part III deals with applications. Its main purpose is to demonstrate how one can find FI's when considering different problems, and then effectively use the general theory exposed in Part II. Perhaps the most illuminating example of the applicability are solutions of long-standing Herstein's conjectures on Lie homomorphisms and Lie derivations - in the proofs practically the entire FI theory is used.

Table of Contents

Preface	p. ix
An Introductory Course	p. 1
What is a Functional Identity?	p. 3
Prerequisites	p. 3
Simple Examples of Functional Identities	p. 8
Basic Concepts	p. 15
Finding Functional Identities in Different Areas	p. 22
The Strong Degree and the FI-Degree	p. 29
The Strong Degree	p. 30
Strongly d-Free Rings and the Fl-Degree	p. 33
Strongly (t; d)-Free Rings	p. 37
s-deg <$>({\cal A})<$> ≤ FI-deg <$>({\cal A})<$>	p. 41
The General Theory	p. 47
Constructing d-Free Sets	p. 49
Notation	p. 49
d-Free Sets	p. 53
Two Constructions of d-Free Sets	p. 58
(t; d)-Free Sets	p. 63
(*;t;d)-Free Rings	p. 71
Functional Identities on d-Free Sets	p. 87
Introducing the General Setting	p. 87
d-Free Pairs	p. 92
Quasi-polynomials and Core Functions	p. 97
Remarks on "Nonlinear" Identities	p. 105
Functional Identities in (Semi)prime Rings	p. 111
The Fractional Degree	p. 111
A List of d-Free Subsets of Prime Rings	p. 119
d-Freeness of Semiprime Rings	p. 123
Commuting Maps on (Semi)prime Rings	p. 128
Generalized Functional Identities	p. 133
Applications	p. 145
Lie Maps and Related Topics	p. 147
Lie Maps on Rings	p. 148
Lie Maps on Skew Elements	p. 158
Lie Maps on Lie Ideals	p. 166
Jordan Maps	p. 174
f-Homomorphisms and f-Derivations	p. 180
Linear Preserver Problems	p. 189
Commutativity Preserving Maps	p. 190
Normality Preserving Maps	p. 201
Zero Jordan Product Preserving Maps	p. 210
Equal Product Preserving Maps	p. 215
Further Applications to Lie Algebras	p. 221
Lie-Admissible Algebras	p. 221
Poisson Algebras	p. 226
Maps Covariant Under the Action of Lie Algebras	p. 230
Maximal Rings of Quotients	p. 235
The Orthogonal Completion	p. 243
Polynomial Identities	p. 249
Generalized Polynomial Identities	p. 255
Bibliography	p. 259
Index	p. 271
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