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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.26315 |
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| _version_ | 1866911548132294656 |
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| author | Garg, Jyoti Maheshwary, Sugandha Setia, Himanshu |
| author_facet | Garg, Jyoti Maheshwary, Sugandha Setia, Himanshu |
| contents | This article determines the structure of the group ring $\mathbb{Z}_nG$, where $G$ is a finite group and $\mathbb{Z}_n$ is the ring of integers modulo $n$, such that $n$ is relatively prime to the order of $G$. The decomposition of $\mathbb{Z}_nG$ is given as a direct sum of matrix rings over Galois rings, thereby extending the structural theory of group rings beyond the classical field setting. We also provide a method to compute a generating set of the unit group $\mathcal{U}(\mathbb{Z}_nG)$, in terms of elementary matrices, using Shoda pair theory. The results are illustrated with examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_26315 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The structure of $\mathbb{Z}_nG$ and its unit group Garg, Jyoti Maheshwary, Sugandha Setia, Himanshu Rings and Algebras This article determines the structure of the group ring $\mathbb{Z}_nG$, where $G$ is a finite group and $\mathbb{Z}_n$ is the ring of integers modulo $n$, such that $n$ is relatively prime to the order of $G$. The decomposition of $\mathbb{Z}_nG$ is given as a direct sum of matrix rings over Galois rings, thereby extending the structural theory of group rings beyond the classical field setting. We also provide a method to compute a generating set of the unit group $\mathcal{U}(\mathbb{Z}_nG)$, in terms of elementary matrices, using Shoda pair theory. The results are illustrated with examples. |
| title | The structure of $\mathbb{Z}_nG$ and its unit group |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2603.26315 |