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 M-n(R) is weakly nil-clean, and to show that the endomorphism ring End(D)(V) over a vector space V-D is weakly nil-clean if and only if it is nil-clean or dim(V) = 1 with D congruent to Z(3).