A note on a pair of derivations of semiprime rings. (Q1777671)

Summary: We study certain properties of derivations on semiprime rings. The main purpose is to prove the following result: let \(R\) be a semiprime ring with center \(Z(R)\), and let \(f\), \(g\) be derivations of \(R\) such that \(f(x)x+xg(x)\in Z(R)\) for all \(x\in R\), then \(f\) and \(g\) are central. As an application, we show that noncommutative semisimple Banach algebras do not admit nonzero linear derivations satisfying the above central property. We also show that every skew-centralizing derivation \(f\) of a semiprime ring \(R\) is skew-commuting.