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

TL;DR

This paper characterizes the structure of the group ring Z_nG for finite groups G and integers n coprime to |G|, and provides a method to compute generators of its unit group.

Contribution

It extends the structural theory of group rings to Z_nG and introduces a new method to compute unit group generators using Shoda pair theory.

Findings

Decomposition of Z_nG into matrix rings over Galois rings.

A method to compute generators of U(Z_nG) using elementary matrices.

Illustrative examples demonstrating the structural results.

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.