On Duo, Reversible and Symmetric Group Rings

Abstract

Let $RG$ denote the group ring of the torsion group $G$ over a commutative ring $R$ with identity. In this paper we present proofs of some statements that appear without to be proved in the literature. We establish the valid implications between the ring-theoretic conditions duo, reversible, SI property and symmetric in the setting of group rings. We further show that if the group ring $RG$ possesses any of these properties, then $G$ is a Hamiltonian group and the characteristic of $R$ is either $0$ or $2$. Moreover, we characterize the same properties in group rings $RG$ in the following cases: ($1)$ $RG$ is a semi-simple group ring and ($2$) $R$ is a semi-simple ring and $G$ any group.