Nil-clean and strongly nil-clean rings

Semi-boolean corner rings

“…Besides, it was obtained in [1,Corollary 7] that if R is a commutative nil-clean ring, then the ring M n (R) is nil-clean. This was extended in [11,Theorem 6.1] to 2-primal strongly nilclean rings and in [11,Corollary 6.8] to strongly nil-clean rings of bounded index of nilpotence. Some more substantial generalizations were established in [6], too.…”

“…It is obvious that abelian (in particular, commutative) nil-clean rings are always strongly nil-clean. But it was independently established in [7] and [11] by exploiting different ideas that a ring R is strongly nil-clean if, and only if, J(R) is nil and R/J(R) is boolean.…”

The strong nil-cleanness of semigroup rings

“…Diesl [1] introduced the concept of nil-clean and strongly nil-clean rings and asked the question when a matrix ring is (strongly) nil-clean. As we have observed, it is difficult to characterize the arbitrary ring R such that ( ) R n is nil-clean [2]. It is also known that the ring of all × 2 2-matrices over any commutative local ring is not strongly nil-clean [3].…”

“…is strongly nil-clean if and only if A B , are strongly nil-clean. Moreover, the strong nilcleanness of Morita contexts, formal matrix rings and generalized matrix rings has also been studied [2]. Noting that semigroup rings are also an extension of group rings, the strong nil-cleanness of certain kinds of group rings has been studied in [2,4,5].…”

“…Note that the term "non-unital ring" or "general ring" refers to a ring that does not necessarily have an identity. It is known that Nicholson extended many of the results on the (strong) cleanness of rings with identity to non-unital rings [7], and that in [1,2], the (strong) nil-cleanness of non-unital rings has been studied.…”