Rings whose elements are sums of m-potents and nilpotents

Rings close to periodic with applications to matrix, endomorphism and group rings

“…(For the partial case of a 2-primal ring, we refer the interested readers for more information to [23] and to [12] as well). Moreover, a similar topic to that in [8] was considered in [59], but unfortunately the proof of [59,Lemma 3.3] is erroneous when the integer m ≥ 2 is not a prime -in fact, the ring Z 4 manifestly demonstrates our claim as simple computations show. However, certain appropriate results from [43] could be applied to avoid these shortcomings.…”

“…is strongly m-nil clean. By considering the denominator, which is obviously a nilpotent ideal, [8,Theorem 1.11] guarantees that E(Z p i k j ) is a strongly m-nil clean ring.…”

“…It was shown in [8,Lemma 3.12] that if R is an m-potent ring for some arbitrary but fixed integer m > 1, that is x m = x for all x ∈ R, then M n (R) is a periodic ring for all naturals n ≥ 1. This assertion was considerably generalized in [11,Corollary 2.22] for any non-fixed m ≥ 2 depending on the element x.…”

“…Some non-trivial examples of such rings can be found in [40] and the bibliography cited therewith. Some related papers of this topic are [8] and [31] as well, which deal with rings whose elements are sums of commutating potent and nilpotent elements.…”

“…Likewise, it was proved in [8,Corollary 3.10] that the matrix ring M n (R) is always nil-clean for all n ≥ 1, provided that the former ring R is weakly 2-primal strongly nilclean -actually, the authors stated this claim without the assumption of being "weakly 2-primal" by using [41,Proposition 2.7] which proposition seems to be false. (For the partial case of a 2-primal ring, we refer the interested readers for more information to [23] and to [12] as well).…”