Rings in which every element is either a sum or a difference of a nilpotent and an idempotent

Abstract

Abstract. Generalizing the notion of nil cleanness from [9], in parallel to [8], we define the concept of weak nil cleanness for an arbitrary ring. Its comprehensive study in different ways is provided as well. A decomposition theorem of a weakly nil-clean ring is obtained. It is completely characterized when an abelian ring is weakly nil-clean. It is also completely determined when a matrix ring over a division ring is weakly nil-clean.

Invo-Clean Unital Rings

“…We define and completely describe the structure of invo-clean rings having identity. We show that these rings are clean but not (weakly) nil-clean and thus they possess independent properties than these obtained by Diesl in [7] and by Breaz-Danchev-Zhou in [2]. …”

“…On the other hand, the latter concept of nil-cleanness was extended in [6] and [2] respectively by defining the notion of weak nil-cleanness as follows: Definition 1.3. A ring R is called weakly nil-clean if every r ∈ R can be presented as either r = q + e or r = q − e, where q ∈ N il(R) and e ∈ Id(R).…”

“…It was established in [2] and [6] that weakly nil-clean rings are themselves clean. Likewise, in [2] was established a complete characterization of abelian weakly nil-clean rings as those abelian rings R for which J(R) is nil and R/J(R) is isomorphic to a Boolean ring B, or to Z 3 , or to B × Z 3 .…”

“…Likewise, in [2] was established a complete characterization of abelian weakly nil-clean rings as those abelian rings R for which J(R) is nil and R/J(R) is isomorphic to a Boolean ring B, or to Z 3 , or to B × Z 3 . (See also [4] for the general case as well as [11] for a slightly different characterization.)…”

“…If R is a nil-clean ring it was proved in [7] that 2 ∈ N il(R), whereas if R is a weakly nil-clean ring it was established in [6] and [2] that 6 ∈ N il(R).…”