On a Functional Equation Related to Jordan Triple Derivations in Prime Rings

Abstract

A classical result of Herstein asserts that any Jordan derivation on a prime ring with char(R) ≠ 2 is a derivation. It is our aim in this paper to prove the following result, which is in the spirit of Herstein’s theorem. Let R be a prime ring with char(R) = 0 or char(R) > 4, and let D: R → R be an additive mapping satisfying the relation D(x⁴) = D(x)x³ + xD(x²)x + x³D(x) for all x ∈ R. In this case, D is a derivation.