Commutative weakly nil clean unital rings

Invo-Clean Unital Rings

“…Also, commutative nil-clean rings are always uniquely nil-clean (compare with [6]), and even it was proved in [7,Corollary 3.8] that strongly nil-clean rings and uniquely strongly nil-clean rings do coincide in general.…”

“…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 .…”

“…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).…”

Weakly Nil-Clean Index and Uniquely Weakly Nil-Clean Rings

“…On the other vein, in [3] and [7] was stated the definition of a weakly nil-clean ring as such a ring R for which any element a ∈ R is of the form a = b + e or a = b − e, where b ∈ N il(R) and e ∈ Id(R). Moreover, a ring R is said to be uniquely weakly nil-clean if the existing idempotent e is unique.…”