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Let R be a prime ring with the extended centroid C and symmetric Martindale quotient ring \(Q_s(R)\). In this paper we prove the following result. Let \(F: R \rightarrow R\) be a generalized derivation associated with a non-zero derivation d on R and let h be an additive map of R such that \(F(x)x=xh(x)\) for all \(x\in R\). Then either R is commutative or \(F(x)=xp\) and \(h(x...

Let be a 2-torsion free ring and let be a noncentral Lie ideal of , and let and be two generalized derivations of . We will analyse the structure of in the following cases: (a) is prime and for all and fixed positive integers ; (b) is prime and for all and fixed integers ; (c) is semiprime and for all and fixed integer ; and (d) is semiprime and for all and fixed integer .