Generalized Derivations and nth Power Maps in Rings

Abstract

We prove that a Jordan (φ, θ)-derivation of a 2-torsion free semiprime ring, with θ an automorphism, must be a (φ, θ)-derivation. This is then used to show more generally that an additive mapping of a semiprime ring to itself that behaves like a generalized (φ, θ)-derivation on nth powers is, in fact, a generalized (φ, θ)-derivation of the ring.