Weakly Clean Rings and Almost Clean Rings

Abstract

Let R be a commutative ring with identity. Nicholson defined R to be clean if each element of R is the sum of a unit and an idempotent. In this paper we study two related classes of rings. We define a ring R to be weakly clean if each element of R can be written as either the sum or difference of a unit and an idempotent and following McGovern we say that R is almost clean if each element of R is the sum of a nonzero-divisor and an idempotent.