m-Power Commuting MAPS on Semiprime Rings
Journal
Communications in Algebra
Journal Volume
42
Journal Issue
3
Pages
1095-1110
Date Issued
2014
Author(s)
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: R → U be an additive m-power commuting map. Suppose that f is Z(R)-linear. It is proved that there exists an idempotent e ∈ C such that ef(x) = λx + μ(x) for all x ∈ R, where λ ∈C and μ: R → C. Moreover, (1 − e)U ≅ M2(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.
Type
journal article