Uniquely Strongly Clean Group Rings (Communications in Mathematical Research, 2021)
A ring $R$ is called clean if every element is the sum of an idempotent anda unit, and $R$ is called uniquely strongly clean (USC for short) if every element isuniquely the sum of an idempotent and a unit that commute. In this article, someconditions on a ring $R$ and a group $G$ such that $RG$ is clean are given. It is alsoshown that if $G$ is a locally finite group, then the group ring $RG$ is USC if and onlyif $R$ is USC, and $G$ is a 2-group. The left uniquely exchange group ring, as a middlering of the uniquely clean ring and the USC ring, does not possess this property, andso does the uniquely exchange group ring.
Keywords: clean ring; group ring; $p$-group; USC ring.