Abstract :
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.
Keywords :
Triangular rings , centralizing maps , k , skew , centralizing maps , nest algebras
