Commutativity of symmetric elements in group rings

Abstract

Let R be a commutative ring with unity and let G be a group. The group ring RG has a natural involution that maps each element of G to its inverse. We denote by RG + the set of symmetric elements under this involution. We study necessary and suffient conditions for RG + to be commutative or, equivalently, for RG + to be a subring of RG . We also determine all torsion groups G such that the set of symmetric units of RG is a subgroup, when char( R ) is an odd prime number.