On weakly nil-clean rings

Abstract

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 {{\displaystyle 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\cong {{\displaystyle Z}}_{\mathrm{3}}.

Key words： Nil-clean ring; strongly nil-clean ring; weakly nil-clean ring; matrix ring; endomorphism ring of a vector space; 2-primal ring

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