Characterization of k-commuting additive maps on rings

Let k be a positive integer and R a ring having unit 1. Denote by Z(R) the center of R. Assume that the characteristic of R is not 2 and there is an idempotent element e∈R such that R satisfies aRe={0}⇒a=0, aR(1−e)={0}⇒a=0, Z(eRe)k=Z(eRe) and Z((1−e)R(1−e))k=Z((1−e)R(1−e)). Then every additive map f:R→R is k-commuting if and only if f(x)=αx+h(x) for all x∈R, where α∈Z(R) and h is an additive map from R into Z(R). As applications, all k-commuting additive maps on prime rings and von Neumann algebras are characterized.