On a functional equation related to derivations in prime rings

Abstract

In this paper we prove the following result. Let m ≥ 1, n ≥ 1 be fixed integers and let R be a prime ring with m + n + 1 ≤ char(R) or char(R) = 0. Suppose there exists an additive nonzero mapping D : R → R satisfying the relation 2D(xn+m+1) = (m + n + 1)(xmD(x)xn + xnD(x)xm) for all \({x\in R}\). In this case R is commutative and D is a derivation.