Uniquely Strongly Clean Group Rings

Abstract

A ring $R$ is called clean if every element is the sum of an idempotent and a unit, and $R$ is called uniquely strongly clean (USC for short) if every element is uniquely the sum of an idempotent and a unit that commute. In this article, some conditions on a ring $R$ and a group $G$ such that $RG$ is clean are given. It is also shown that if $G$ is a locally finite group, then the group ring $RG$ is USC if and only if $R$ is USC, and $G$ is a 2-group. The left uniquely exchange group ring, as a middle ring of the uniquely clean ring and the USC ring, does not possess this property, and so does the uniquely exchange group ring.