Some results concerning n-σ-centralizing mappings in semiprime rings

Let n ≥ 1 be a fixed integer. Let R be a semiprime ring and S an additive subgroup of R, σ, τ two endomorphisms of R and F: R→ R an additive mapping of R. In the present paper, we prove that (1)if R is (n + 1)!-torsion free, S is (n + 1)-power closed and [F(x) , σ(x) n] ∈ Z(R) for all x∈ S, then [F(x) , σ(x) n] = 0 for all x∈ S;(2) if R is 3!-torsion free, S is square closed and [[F(x) , σ(x)] , σ(x)] ∈ Z(R) for all x∈ S, then [[F(x) , σ(x)] , σ(x)] = 0 for all x∈ S. We also consider a number of applications in semiprime rings with derivations, (σ, τ)-derivations and generalized derivations, and extend some known results in the literature. [MediaObject not available: see fulltext.] © 2014, The Author(s).