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arXiv · 2501.02072

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

Kisnney Emiliano de Almeida·Jacqueline Costa Cintra·Mauricio Araujo Ferreira·Edward Landi Tonucci
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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 $Q_8$.

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Kisnney Emiliano de Almeida, Jacqueline Costa Cintra, Mauricio Araujo Ferreira, Edward Landi Tonucci. 2025-01-03. On $*$-Clean Group Rings over SLC-groups. https://arxiv.org/abs/2501.02072

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