Some results concerning additive mappings and derivations on semiprime rings

Abstract

Let R be a 2-torsion free semiprime ring and let f : R ! R be an additive mapping satisfying the relation [f(x); x2] = 0 for all x 2 R. We prove that in this case [f(x); x] = 0 holds for all x 2 R. This result makes it possible to prove the following result. Let R be a 2-torsion free semiprime ring and let D;G : R ! R be derivations. Suppose that the relation [D2(x) + G(x); x2] = 0 holds for all x 2 R. Then D and G both map R into its center.