Bresar / Bre#ar / Chebotar Functional Identities

2007
ISBN: 978-3-7643-7796-0
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 272 Seiten

Reihe: Frontiers in Mathematics

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.

Autoren/Hrsg.

Bresar, Matej

Bre#ar, Matej

Chebotar, Mikhail A.

Martindale, Wallace S.

Inhaltsverzeichnis

1;Contents;7
2;Preface;9
3;I An Introductory Course;13
3.1;What is a Functional Identity?;14
3.1.1;1.1 Prerequisites;14
3.1.2;1.2 Simple Examples of Functional Identities;19
3.1.3;1.3 Basic Concepts;26
3.1.4;1.4 Finding Functional Identities in Different Areas;33
3.2;The Strong Degree and the FI- Degree;40
3.2.1;2.1 The Strong Degree;41
3.2.2;2.2 Strongly d-Free Rings and the FI-Degree;44
3.2.3;2.3 Strongly (t; d)-Free Rings;48
3.2.4;2.4 s-deg(A) = FI-deg(A);52
4;II The General Theory;57
4.1;Constructing d-Free Sets;58
4.1.1;3.1 Notation;58
4.1.2;3.2 d-Free Sets;62
4.1.3;3.3 Two Constructions of d-Free Sets;67
4.1.4;3.4 (t; d)-Free Sets;72
4.1.5;3.5 (*; t; d)-Free Rings;80
4.2;Functional Identities on d-Free Sets;95
4.2.1;4.1 Introducing the General Setting;95
4.2.2;4.2 d-Free Pairs;100
4.2.3;4.3 Quasi-polynomials and Core Functions;105
4.2.4;4.4 Remarks on Nonlinear” Identities;113
4.3;Functional Identities in ( Semi) prime Rings;119
4.3.1;5.1 The Fractional Degree;119
4.3.2;5.2 A List of d-Free Subsets of Prime Rings;127
4.3.3;5.3 d-Freeness of Semiprime Rings;131
4.3.4;5.4 Commuting Maps on (Semi)prime Rings;136
4.3.5;5.5 Generalized Functional Identities;141
5;III Applications;153
5.1;Lie Maps and Related Topics;154
5.1.1;6.1 Lie Maps on Rings;155
5.1.2;6.2 Lie Maps on Skew Elements;165
5.1.3;6.3 Lie Maps on Lie Ideals;173
5.1.4;6.4 Jordan Maps;181
5.1.5;6.5 f-Homomorphisms and f-Derivations;187
5.2;Linear Preserver Problems;196
5.2.1;7.1 Commutativity Preserving Maps;197
5.2.2;7.2 Normality Preserving Maps;208
5.2.3;7.3 Zero Jordan Product Preserving Maps;217
5.2.4;7.4 Equal Product Preserving Maps;222
5.3;Further Applications to Lie Algebras;227
5.3.1;8.1 Lie-Admissible Algebras;227
5.3.2;8.2 Poisson Algebras;232
5.3.3;8.3 Maps Covariant Under the Action of Lie Algebras;236
6;Appendices;240
6.1;Maximal Rings of Quotients;240
6.2;The Orthogonal Completion;247
6.3;Polynomial Identities;252
6.4;Generalized Polynomial Identities;257
7;Bibliography;261
8;Index;273