On θ-centralizers of semiprime rings

Let R be a semiprime ring with center Z(R) and µ be a surjective homomorphism. In this paper, we prove that T is a µ-centralizer if one of the following holds: (i) T(x)µ(y) = µ(x)T(y) for all x;y 2 R, where T is a mapping. (ii) [T(x);µ(x)] = 0 for all x 2 R, where T is a left µ-centralizer. (iii) 2T(xyx) = T(x)µ(y)µ(x) + µ(x)µ(y)T(x) for all x;y 2 R, where T is an additive mapping, µ(Z(R)) = Z(R) and R is 2-torsion free. 1