On Duo, Reversible and Symmetric Group Rings

TL;DR

This paper investigates the properties of group rings over torsion groups, establishing implications between ring-theoretic conditions and characterizing when these properties hold based on the structure of the group and the ring.

Contribution

It provides proofs of known but unproven statements about duo, reversible, SI, and symmetric properties in group rings, and characterizes these properties for semi-simple cases.

Findings

Group rings with these properties imply the group is Hamiltonian.

The characteristic of the ring is 0 or 2 when properties hold.

Characterizations for semi-simple group rings and 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.