On the unit group of a commutative group ring

TL;DR

This paper studies the structure of the group of units in the group algebra over a finite abelian p-group with coefficients in a residue ring, extending understanding of algebraic units in modular group rings.

Contribution

It provides new insights into the structure of the unit group of the group algebra over rings of residues modulo p^e for finite abelian p-groups.

Findings

Characterization of the unit group structure

Identification of generators and relations

Extension of known results to higher powers of p

Abstract

We investigate the group of normalized units of the group algebra $\mathbb{Z}_{p^e}G$ of a finite abelian $p$-group $G$ over the ring $\mathbb{Z}_{p^e}$ of residues modulo $p^e$ with $e\geq 1$.