Group Identities on Units and Symmetric Units of Group Rings

  • Gregory T.┬áLee
Part of the Algebra and Applications book series (AA, volume 12)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Gregory T. Lee
    Pages 1-43
  3. Gregory T. Lee
    Pages 45-75
  4. Gregory T. Lee
    Pages 77-101
  5. Gregory T. Lee
    Pages 137-147
  6. Gregory T. Lee
    Pages 161-169
  7. Back Matter
    Pages 171-194

About this book

Introduction

Let FG be the group ring of a group G over a field F. Write U(FG) for the group of units of FG. It is an important problem to determine the conditions under which U(FG) satisfies a group identity. In the mid 1990s, a conjecture of Hartley was verified, namely, if U(FG) satisfies a group identity, and G is torsion, then FG satisfies a polynomial identity. Necessary and sufficient conditions for U(FG) to satisfy a group identity soon followed.

Since the late 1990s, many papers have been devoted to the study of the symmetric units; that is, those units u satisfying u* = u, where * is the involution on FG defined by sending each element of G to its inverse. The conditions under which these symmetric units satisfy a group identity have now been determined.

This book presents these results for arbitrary group identities, as well as the conditions under which the unit group or the set of symmetric units satisfies several particular group identities of interest.

Keywords

Group identities Group rings Involutions Lie Prime Symmetric elements Units field group identity polynomial prime number ring set torsion

Authors and affiliations

  • Gregory T.┬áLee
    • 1
  1. 1., Department of Mathematical SciencesLakehead UniversityThunder BayCanada

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-84996-504-0
  • Copyright Information Springer-Verlag London Limited 2010
  • Publisher Name Springer, London
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-84996-503-3
  • Online ISBN 978-1-84996-504-0