On Certain Identity Related to Herstein Theorem on Jordan Derivations

Abstract

Let R be a 6-torsion-free prime ring and let \({D : R \rightarrow R}\) be an additive mapping satisfying the relation 2D(x4) = D(x3)x + x3D(x) + D(x)x3 + xD(x3) for all \({x \in R}\) . The purpose of this paper is to show that D is a derivation. This result is related to a classical result of Herstein, which states that any Jordan derivation on a 2-torsion-free prime ring is a derivation.