(untitled) (Communications in Mathematical Research, 2021)
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.