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

Let A and B be unital ringsý, ýand Mý ýbe an (Aý,ýB)-bimoduleý, ýwhich is faithful as aý ýleft A-module and also as a right B-moduleý. ýLet U=\rm Tri(Aý,ýMý,ýýýB) be the triangular ring and Z(U) itsý ýcenterý. ýAssume that f:U→U is a mapý ýsatisfying f(x)−f(x)−f(y)∈Z(U) for allý ýx, y∈U and k is a positive integerý. ýIt is showný ýthatý, ýunder some mild conditionsý, ýthe following statements areý ýequivalentý: ý(1) [f(x),xk]∈Z(U) for allý ýx∈U; (2) [f(x),xk]=0 for all x∈U;ý ý(3) [f(x),x]=0 for all x∈U; (4) there exist aý ýcentral element z∈Z(U) and an additiveý ýmodulo Z(U) map h:ýýU→Z(U) such that f(x)=zx(x)ý ýfor all x∈Uý. ýIt is also shown that there is noý ýnonzero additive k-skew-centralizing maps on triangular rings.