Functional Identities (Frontiers in Mathematics)

✍ Scribed by Matej Brear, Mikhail A. Chebotar, Wallace S. Martindale

• Publisher: Birkhäuser Basel
• Year: 2007
• Tongue: English
• Leaves: 284
• Edition: 1
• Category: Library

✦ Synopsis

A functional identity can be informally described as an identical relation involving arbitrary elements in an associative ring together with arbitrary (unknown) functions. The theory of functional identities is a relatively new one, and this is the first book on this subject. The book is accessible to a wide audience and touches on a variety of mathematical areas such as ring theory, algebra and operator theory.

✦ Table of Contents

Contents......Page 7
Preface......Page 9
1.1 Prerequisites......Page 15
1.2 Simple Examples of Functional Identities......Page 20
1.3 Basic Concepts......Page 27
1.4 Finding Functional Identities in Di.erent Areas......Page 34
2 The Strong Degree and the FI-Degree......Page 41
2.1 The Strong Degree......Page 42
2.2 Strongly d-Free Rings and the FI-Degree......Page 45
2.3 Strongly (t; d)-Free Rings......Page 49
2.4 s-deg(A) ≤ FI-deg(A)......Page 53
3.1 Notation......Page 61
3.2 d-Free Sets......Page 65
3.3 Two Constructions of d-Free Sets......Page 70
3.4 (t; d)-Free Sets......Page 75
3.5 (∗; t; d)-Free Rings......Page 83
4.1 Introducing the General Setting......Page 99
4.2 d-Free Pairs......Page 104
4.3 Quasi-polynomials and Core Functions......Page 109
4.4 Remarks on “Nonlinear” Identities......Page 117
5.1 The Fractional Degree......Page 123
5.2 A List of d-Free Subsets of Prime Rings......Page 131
5.3 d-Freeness of Semiprime Rings......Page 135
5.4 Commuting Maps on (Semi)prime Rings......Page 140
5.5 Generalized Functional Identities......Page 145
6 Lie Maps and Related Topics......Page 159
6.1 Lie Maps on Rings......Page 160
6.2 Lie Maps on Skew Elements......Page 170
6.3 Lie Maps on Lie Ideals......Page 178
6.4 Jordan Maps......Page 186
6.5 f-Homomorphisms and f-Derivations......Page 192
7 Linear Preserver Problems......Page 201
7.1 Commutativity Preserving Maps......Page 202
7.2 Normality Preserving Maps......Page 213
7.3 Zero Jordan Product Preserving Maps......Page 222
7.4 Equal Product Preserving Maps......Page 227
8.1 Lie-Admissible Algebras......Page 233
8.2 Poisson Algebras......Page 238
8.3 Maps Covariant Under the Action of Lie Algebras......Page 242
Appendix A: Maximal Rings of Quotients......Page 247
Appendix B: The Orthogonal Completion......Page 255
Appendix C: Polynomial Identities......Page 261
Appendix D: Generalized Polynomial Identities......Page 267
Bibliography......Page 271
Index......Page 283