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arXiv · 2603.26315

The structure of $\mathbb{Z}_nG$ and its unit group

Jyoti Garg·Sugandha Maheshwary·Himanshu Setia
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Abstract

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.

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Jyoti Garg, Sugandha Maheshwary, Himanshu Setia. 2026-03-27. The structure of $\mathbb{Z}_nG$ and its unit group. https://arxiv.org/abs/2603.26315

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