A note on weakly clean rings

Abstract

Let R be an associative ring with identity. An element x ∈ R is said to be weakly clean if x = u + e or x = u − e for some unit u and idempotent e in R. The ring R is said to be weakly clean if all of its elements are weakly clean. In this paper we obtain an element-wise characterization of abelian weakly clean rings. A relation between unit regular rings and weakly clean rings is also obtained.

When Is an Abelian Weakly Clean Ring Clean?

“…We first begin with the promised above extension of Theorem 2.1 from [5], especially of the implication (⇒), in the sense of Definition 2.1 in [3]. Theorem 2.1.…”

“…The sufficiency follows directly from [5], so that we will concentrate on the necessity. To that aim, given x ∈ R, then we may write x = u + f or x = u − f with u ∈ U (R) and f ∈ Id(R).…”

“…It is worthwhile notice that in [1] was constructed a concrete example of a commutative weakly clean ring in which 2 ∈ U (R) that is not clean; for instance, this is the ring Z (3) ∩ Z (5) , consisting of the joint elements of Z (3) and Z (5) , where Z (3) is the ring of integers localized at 3 and Z (5) is the ring of integers localized at 5.…”

“…Moreover, the next necessary and sufficient condition for an abelian ring to be weakly clean, slightly reformulated for the left hand-side, is needed for our further applicable purposes (see [5] or [12]):…”

On weakly clean and weakly exchange rings having the strong property

“…On the other hand, by virtue of [20] or [24], a ring R is an exchange ring if, for every x ∈ R, there exists an idempotent e such that e ∈ xR and 1 − e ∈ (1 − x)R. It is well known that clean rings are exchange while the converse is untrue; for abelian rings these two sorts of rings, however, coincide. Generalizing this, in [29] (see [6] or [7] too) a ring R is said to be weakly exchange if, for each x ∈ R, there exists an idempotent e such that e ∈ xR and 1 − e ∈ (1 − x)R or 1 − e ∈ (1 + x)R. By the proof of the necessity of Theorem 2.1 from [6], it is well known that weakly clean rings are in general weakly exchange, but the converse manifestly fails. However, for abelian rings, in the cited theorem from [6] was shown that weakly exchange rings are precisely the weakly clean rings.…”