k-power centralizing and k-power skew-centralizing maps on‎ ‎triangular rings

Let A and B be unital rings‎, ‎and M‎ ‎be an (A‎,‎B)-bimodule‎, ‎which is faithful as a‎ ‎left A-module and also as a right B-module‎. ‎Let U=Tri(A‎,‎M‎,‎‎‎B) be the triangular ring and Z(U) its‎ ‎center‎. ‎Assume that f:U→U is a map‎ ‎satisfying f(x+y)−f(x)−f(y)∈Z(U) for all‎ ‎x, y∈U and k is a positive integer‎. ‎It is shown‎ ‎that‎, ‎under some mild conditions‎, ‎the following statements are‎ ‎equivalent‎: ‎(1) [f(x),x^k]∈Z(U) for all‎ ‎x∈U; (2) [f(x),x^k]=0 for all x∈U;‎ ‎(3) [f(x),x]=0 for all x∈U; (4) there exist a‎ ‎central element z∈Z(U) and an additive‎ ‎modulo Z(U) map h:‎‎U→Z(U) such that f(x)=zx+h(x)‎ ‎for all x∈U‎. ‎It is also shown that there is no‎ ‎nonzero additive k-skew-centralizing maps on triangular rings.