Commutative rings whose elements are a sum of a unit and idempotent

Abstract

As defined by Nicholson a (noncommutative) ring R is a clean ring if every element of R is a sum of a unit and an idempotent. Let R be a commutative ring with identity. We define R to be a uniquely clean ring if every element of R can be written uniquely as the sum of a unit and an idempotent. Examples of clean rings (uniquely clean rings) include von Neumann regular rings (Boolean rings) and quasilocal rings (with residue field ℤ2). A ring R is a clean ring or uniquely clean ring if and only if R/√0 is. So every zero-dimensional ring R is a clean ring, but a zero-dimensional ring R is a uniquely clean ring if and only if R/√0 is a Boolean ring.