On weakly nil-clean rings.

We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of an abelian weakly nil-clean ring, and of a strongly nil-clean ring. As applications, this result is used to determine the 2-primal rings R such that the matrix ring $$\mathbb{M}_n (R)$$ is weakly nil-clean, and to show that the endomorphism ring End( V) over a vector space V is weakly nil-clean if and only if it is nil-clean or dim( V) = 1 with D≅= ℤ. [ABSTRACT FROM AUTHOR]