Let \(\mathcal {R}\) be a prime ring with \(char(\mathcal {R})\ne 2\) , \(\mathcal {U}\) the Utumi quotient ring of \(\mathcal {R}\) and \(\mathcal {C}=\mathcal {Z}(\mathcal {U})\) the extended centroid of \(\mathcal {R}\) and \(\mathcal {L}\) a non-central Lie ideal of \(\mathcal {R}\) . If \(\mathcal {F}\) , \(\mathcal {G}\) and \(\mathcal {H}\) are three generalized derivations of \(\mathcal {R}\) such that \(\begin{aligned} {[}\mathcal {F}(X)\mathcal {G}(X)-X \mathcal {H}(X),X^{t_1},X^{t_2},\ldots ,X^{t_n}{]}=0 \end{aligned}\) for all \(X \in \mathcal {L}\) and for some fixed positive integers \(t_1,\ldots ,t_n\) , then the complete characterization of the maps \(\mathcal {F}\) , \(\mathcal {G}\) and \(\mathcal {H}\) are described.
