On generalized derivations satisfying certain identities - Albaş, E. (Ukrains’kyi Matematychnyi Zhurnal, 2011)
Let $R$ be a prime ring with char $R eq 2$ and $d$ be a generalized derivation on $R$. The goal of this studyis to investigate the generalized derivation $d$ satisfying any one of the following identities:$$(i) \quad d[(x, y)] = [d(x), d(y)] \quad \text{for all} x, y \in R;$$$$(ii) \quad d[(x, y)] = [d(y), d(x)] \quad \text{for all} x, y \in R;$$$$(iii)\quad d([x, y]) = [d(x), d(y)] \text{either} d([x, y]) = [d(y), d(x)] \quad \text{for all} x, y \in R$$.