m-POWER COMMUTING MAPS ON SEMIPRIME RINGS

Abstract

Let R be a semiprime ring with center Z(R), extended centroid C, U the maximal right ring of quotients of R, and m a positive integer. Let f: RU be an additive m-power commuting map. Suppose that f is Z(R)-linear. It is proved that there exists an idempotent eC such that ef(x)=x+(x) for all xR, where C and : RC. Moreover, (1-e)UM2(E), where E is a complete Boolean ring. As consequences of the theorem, it is proved that every additive, 2-power commuting map or centralizing map from R to U is commuting.