Identity on theta-centralizers in semiprime rings

Abstract

Let m and n be fixed integers with m ≥ 1, n ≥ 1 and let R be a (m + n +2)!-torsion free semiprime ring with the identity element suppose there exists an additive mapping T: R → R such that T(x^m+n+1) = θ(x^m) T(x) θ(xⁿ) holds for all x ∈ R where θ is an automorphisms of R. In this case T is a centralizer.