On the identity \(D³+D²=G\) in semiprime rings

The purpose of this paper is to prove the following result. Let \(R\) be a \(2-\)torsion free semiprime ring and let \(D, G : R → R\) be derivations. Suppose that \(D³(x)+D²(x)=G(x)\) holds for all \(x ∈ R\). In this case \(D=0\). The result is related to the classical theory of derivations initiated by Posner.