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On the automorphisms of the group ring of a unique product group

Published online by Cambridge University Press:  20 January 2009

A. A. Mehrvarz  and
D. A. R. Wallace
A. A. Mehrvarz
Affiliation:
Department of Mathematics, University of Tabriz, Tabriz, Iran
D. A. R. Wallace
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow Gl 1XH, Scotland
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Let R be a ring with an identity and a nilpotent ideal N. Let G be a group and let R(G) be the group ring of G over R. The aim of this paper is to study the relationships between the automorphisms of G and R-linear automorphisms of R(G) which either preserve the augmentation or do so modulo the ideal N. We shall show, for example, that if G is a unique product group ([6], Chapter 13, Section 1) then every automorphism of R(G) is modulo N induced from some automorphism of G. This result, which is immediate if, for instance, R is an integral domain, is here requiring of proof since R(G) has non-trivial units (e.g. if N ≠ 0, 1 + n(g − h), ∀ n ∈ N, ∀g, h ∈ G is a unit of augmentation 1), the existence of which is responsible for some of the difficulties inherent in the present investigation. We are obliged to the referee for several helpful suggestions and, in particular, for the proof of Lemma 2.2 whose use obviates our previous combinatorial arguments.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 30 , Issue 2 , June 1987 , pp. 201 - 205
Copyright
Copyright © Edinburgh Mathematical Society 1987

References

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