On some equations related to derivations in rings

Abstract

Let m and n be positive integers with m+n≠0, and let R be an (m+n+2)!-torsion free semiprime ring with identity element. Suppose there exists an additive mapping D:R→R, such that D(xm+n+1)=(m+n+1)xmD(x)xn is fulfilled for all x∈R, then D is a derivation which maps R into its center.