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

TL;DR

This paper analyzes when the group ring $RG$ is $*$-clean for SLC-groups $G$ equipped with the canonical involution. It develops general transfer principles for $*$-cleanness, provides a detailed structural restriction showing that $G$ must be $Q_8\times A$ with $A$ abelian and constrained torsion, and proves a no-solution condition on a sum-of-squares equation that acts as both a necessary and, in certain cases, a sufficient criterion. The results extend known $Q_8$-based findings to product groups with elementary abelian $2$-groups and connect $*$-cleanness to cleanness through a decomposition of group rings, yielding concrete criteria for semisimple coefficient rings and fields via sums of squares in cyclotomic extensions. These insights yield explicit corollaries for primes modulo $8$ and provide a framework to assess $*$-cleanness in a broad family of group rings arising from SLC-groups, with potential implications for related involution-based properties in group rings.

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 $Q_8$.

Theorems & Definitions (39)

• Lemma 2.1
• proof
• Proposition 2.2
• Proposition 2.3
• Lemma 2.4
• proof
• Proposition 2.5
• proof
• Proposition 2.6
• Theorem 2.7