The structure of ${\mathbb{Z}}_{n}G$ and its unit group

Abstract

This article determines the structure of the group ring ${\mathbb{Z}}_{n}G$, 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 ${\mathbb{Z}}_{n}G$ 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 $\mathcal{U}({\mathbb{Z}}_{n}G)$, in terms of elementary matrices, using Shoda pair theory. The results are illustrated with examples.