Clean Index of Rings

Abstract

A ring is called clean (resp., uniquely clean) if each of its elements can be (resp., uniquely) expressed as the sum of an idempotent and a unit. Motivated by recent work on uniquely clean rings in [6], we introduce the clean index of a ring R. For a ∈ R, let ℰ(a) = {e ∈ R: e 2 = e, a − e ∈ U(R)} where U(R) is the group of units of R and the clean index of R, denoted in(R), is defined by in(R) = sup{|ℰ(a)|: a ∈ R}. Thus, R is uniquely clean if and only if R is clean with in(R) = 1. So far, uniquely clean rings are the only clean rings whose structure is fully understood (see [6]). In this article, we characterize the (arbitrary) rings of clean indices 1, 2, 3 and determine the abelian rings of finite clean index. Applications to semipotent rings, semiprime rings, and clean rings are discussed.