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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.02072 |
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| _version_ | 1866912176715857920 |
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| author | de Almeida, Kisnney Emiliano Cintra, Jacqueline Costa Ferreira, Mauricio Araujo Tonucci, Edward Landi |
| author_facet | de Almeida, Kisnney Emiliano Cintra, Jacqueline Costa Ferreira, Mauricio Araujo Tonucci, Edward Landi |
| contents | 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 $Q_8$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_02072 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $*$-Clean Group Rings over SLC-groups de Almeida, Kisnney Emiliano Cintra, Jacqueline Costa Ferreira, Mauricio Araujo Tonucci, Edward Landi Rings and Algebras 16S34, 13F99, 20E99 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 $Q_8$. |
| title | On $*$-Clean Group Rings over SLC-groups |
| topic | Rings and Algebras 16S34, 13F99, 20E99 |
| url | https://arxiv.org/abs/2501.02072 |