On $*$-Clean Group Rings over SLC-groups

TL;DR

This paper investigates the $*$-cleanness property in group rings over SLC-groups with their canonical involution, characterizing when such rings are $*$-clean and generalizing previous results for the quaternion group.

Contribution

It establishes conditions under which the group ring over an SLC-group is $*$-clean, specifically linking $*$-cleanness to the group being a product involving $Q_8$, and extends prior work to broader cases.

Findings

If $RG$ is $*$-clean, then $G$ is a product of $Q_8$ and an abelian group.

Provides a partial converse for specific cases, extending Gao, Chen, and Li's results.

Generalizes known results about $*$-clean group rings over $Q_8$.

Abstract

The property of $*$-cleanness in group rings has been studied for some groups considering the classical involution, given by $g^*=g^{-1}$. A group is called an SLC-group if its quotient by its center is isomorphic to the Klein group; these groups are equipped with its own canonical involution, which usually does not coincide with the classical one. In this paper we study the $*$-cleanness of $RG$ when $G$ is an SLC-group, considering $*$ as its canonical involution. In that context, we prove that if $RG$ is $*$-clean then $G$ is the direct product of $Q_8$ and an abelian group with some extra properties and we find a converse for some specific cases, generalizing a result by Gao, Chen and Li for $Q_8$.