Trivial units in commutative group rings of \(G\times C_{n}\)
It is known that if the unit group of an integral group ring \(\mathbb{Z}G\) is trivial, then the unit group of \(\mathbb{Z}(G\times C_{2})\) is trivial as well [3]. The aim of this study is twofold: firstly, to identify rings \(R\) that are \(D\)-adapted for the direct product \(D=G\times H\) of ab...
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| Datum: | 2024 |
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Lugansk National Taras Shevchenko University
2024
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2086 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543301990088704 |
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| author | Küsmüş, Ömer |
| author_facet | Küsmüş, Ömer |
| author_sort | Küsmüş, Ömer |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2024-06-27T08:42:43Z |
| description | It is known that if the unit group of an integral group ring \(\mathbb{Z}G\) is trivial, then the unit group of \(\mathbb{Z}(G\times C_{2})\) is trivial as well [3]. The aim of this study is twofold: firstly, to identify rings \(R\) that are \(D\)-adapted for the direct product \(D=G\times H\) of abelian groups \(G\) and \(H\), such that the unit group of the ring \(R(G\times H)\) is trivial. Our second objective is to investigate the necessary and sufficient conditions on both the ring \(R\) and the direct factors of \(D\) to satisfy the property that the normalized unit group \(V(RD)\) is trivial in the case where \(D\) is one of the groups \(G\times C_{3}\), \(G\times K_{4}\) or \(G\times C_{4}\), where \(G\) is an arbitrary finite abelian group, \(C_{n}\) denotes a cyclic group of order \(n\) and \(K_{4}\) is Klein 4-group. Hence, the study extends the related result in [18]. |
| first_indexed | 2025-12-02T15:26:07Z |
| format | Article |
| id | admjournalluguniveduua-article-2086 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:26:07Z |
| publishDate | 2024 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-20862024-06-27T08:42:43Z Trivial units in commutative group rings of \(G\times C_{n}\) Küsmüş, Ömer trivial units, commutative, group rings, direct product 16S34, 16U60 It is known that if the unit group of an integral group ring \(\mathbb{Z}G\) is trivial, then the unit group of \(\mathbb{Z}(G\times C_{2})\) is trivial as well [3]. The aim of this study is twofold: firstly, to identify rings \(R\) that are \(D\)-adapted for the direct product \(D=G\times H\) of abelian groups \(G\) and \(H\), such that the unit group of the ring \(R(G\times H)\) is trivial. Our second objective is to investigate the necessary and sufficient conditions on both the ring \(R\) and the direct factors of \(D\) to satisfy the property that the normalized unit group \(V(RD)\) is trivial in the case where \(D\) is one of the groups \(G\times C_{3}\), \(G\times K_{4}\) or \(G\times C_{4}\), where \(G\) is an arbitrary finite abelian group, \(C_{n}\) denotes a cyclic group of order \(n\) and \(K_{4}\) is Klein 4-group. Hence, the study extends the related result in [18]. Lugansk National Taras Shevchenko University 2024-06-27 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2086 10.12958/adm2086 Algebra and Discrete Mathematics; Vol 37, No 2 (2024) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2086/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2086/1072 Copyright (c) 2024 Algebra and Discrete Mathematics |
| spellingShingle | trivial units commutative group rings direct product 16S34 16U60 Küsmüş, Ömer Trivial units in commutative group rings of \(G\times C_{n}\) |
| title | Trivial units in commutative group rings of \(G\times C_{n}\) |
| title_full | Trivial units in commutative group rings of \(G\times C_{n}\) |
| title_fullStr | Trivial units in commutative group rings of \(G\times C_{n}\) |
| title_full_unstemmed | Trivial units in commutative group rings of \(G\times C_{n}\) |
| title_short | Trivial units in commutative group rings of \(G\times C_{n}\) |
| title_sort | trivial units in commutative group rings of \(g\times c_{n}\) |
| topic | trivial units commutative group rings direct product 16S34 16U60 |
| topic_facet | trivial units commutative group rings direct product 16S34 16U60 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2086 |
| work_keys_str_mv | AT kusmusomer trivialunitsincommutativegroupringsofgtimescn |