Symmetric Elements Under Oriented Involutions in Group Rings

Abstract

Let RG denote the group ring of a group G over a commutative ring with unity R. Given a homomorphism σ:G → {± 1} and an involution ϕ of the group G, an oriented invoultion ^σϕ of RG is defined in a natural way. We characterize when the set of symmetric elements under this involution is a subring. This gives a unified setting for earlier work of several authors.

Key Words:

• Group ring
• Oriented involution
• Symmetric elements
• Symmetric units

2000 Mathematics Subject Classification:

• Primary 16U60
• Secondary 16W10, 20C07

ACKNOWLEDGEMENT

The authors are grateful to the referee for comments and suggestions that helped to improve the final version of this article. This work was partially supported by Fapesp, Procs. 03/07246-3 and 2000/07291-0 and CNPq 300243/79-0(RN).

Notes

Communicated by S. Sehgal.