Symmetric Elements of Nonlinear Involutions in Group Rings

Abstract

Given an involution φ : G → G in a group G and a ring R, we study the extensions, not necessarily linear, to an involution ψ : RG → RG in the group ring RG. We investigate the symmetric elements, those α ∈ RG for which ψ(α) = α, and give necessary and sufficient conditions for the set of symmetric elements, (RG)ψ, to be a subring of RG. This work is a generalization of [6] and references therein where only linear extensions of the group involution are considered.