On the derivations of semiprime rings and noncommutative Banach algebras (Q1569952)

From the study of derivations on semiprime rings a generalization of Jordan derivations on noncommutative Banach algebras is developed. The main theorem is: Let \(D\) be a continuous linear Jordan derivation \(D\colon A\to A\) such that \([D(x),x]D(x)[D(x),x]\in\text{rad }A\) for all \(x\in A\). This implies \(D(A)\subseteq\text{rad }A\).