A result concerning derivations in prime rings. (Q2843830)

Let \(R\) be a prime ring and \(D\colon R\to R\) an additive map. The authors consider two identities involving \(D\) that are similar to the defining property of a Jordan derivation, and they show that then \(D\) must be a derivation of \(R\). Specifically, they prove that if \(D(x^3)=D(x^2)x+x^2D(x)\) for all \(x\in R\), or if \(D(x^3)=D(x)x^2+xD(x^2)\) for all \(x\in R\), then \(D\) is a derivation of \(R\) when \(\text{char\,}R\neq 2,3\).