R-Clean Group Rings

Abstract

Let R be an associative ring with unity. An element a ∈ R is said to be r-clean if a = e + r, where e is an idempotent and r is a von Neumann regular element in R. If every element of R is r-clean, then R is called an r-clean ring. In this paper, we investigate the conditions under which the group ring RG is r-clean. We show that if R is a ring and G is a locally finite p-group with p ∈ J(R), then the group ring RG is r-clean if and only if R is r-clean.