A Note on Weakly Nil-Clean Rings.

A ring R is (strongly) weakly nil clean if every element in R is the sum or difference of a nilpotent and an idempotent (that commutes). In this note, we show that if R is strongly nil clean such that J(R) is locally nilpotent, then M n (R) is weakly nil clean. We also give a characterization of strongly weakly nil cleanness of the group ring RG where R is a ring and G is a group, and a characterization of weakly nil cleanness of the group ring RG, when R is a ring and G is a nilpotent group. If A and B are two strongly weakly nil-clean k-algebras (k is a commutative ring) such that J(A) and J(B) are locally nilpotent, then A ⊗ k B is a strongly weakly nil-clean k-algebra. This gives an answer to the question posed in [21]. [ABSTRACT FROM AUTHOR]