Functional Identities (eBook)

Matej Brešar, Mikhail A. Chebotar, Wallace S. Martindale (Autoren)

eBook Download: PDF

2007 | 2007
XII, 272 Seiten
Springer Basel (Verlag)
978-3-7643-7796-0 (ISBN)

Infos

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.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 8.8.2007
• Reihe/Serie: Frontiers in Mathematics
• Reihe/Serie: Frontiers in Mathematics
• Zusatzinfo: XII, 272 p.
• Verlagsort: Basel
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Technik
• Schlagworte: Algebra • functional identity • Homomorphism • Lie algebra • linear algebra • operator theory • Ring Theory
• ISBN-10: 3-7643-7796-8 / 3764377968
• ISBN-13: 978-3-7643-7796-0 / 9783764377960