Complementation in the Group of Units of Matrix Rings

TL;DR

This paper investigates the conditions under which the subgroup of units of the form 1 plus the Jacobson radical has a complement in the group of units of matrix rings over certain rings, completing previous partial results.

Contribution

It determines the existence of complements in the group of units for matrix rings over rings of the form Z_{p^k}, extending prior work by Coleman and Easdown.

Findings

Complements exist for certain n, p, k configurations.

Complements do not exist in some cases, fully characterizing the scenarios.

Provides a complete classification for matrix rings over Z_{p^k}.

Abstract

Let $R$ be a ring with $1$ and $\J(R)$ its Jacobson radical. Then $1+\J(R)$ is a normal subgroup of the group of units, $G(R)$. The existence of a complement to this subgroup was explored in a paper by Coleman and Easdown; in particular the ring $R=\Mat_n(\Z_{p^k})$ was considered. We prove the remaining cases to determine for which $n$, $p$ and $k$ a complement exists in this ring.