Hyper-commuting generalized derivations acting on Lie ideals in prime rings

Abstract

UDC 512.552

Let $\mathscr{R}$ be a prime ring with its Utumi ring of quotients $\mathscr{U}$ and extended centroid $\mathcal{C}.$ Also let $n\geq 1$ be a fixed integer and let ${\rm char}(\mathscr{R})\neq 2.$ Suppose that $\mathscr{F}, \mathcal{G},$ and $\mathscr{H}$ are three generalized derivations of $\mathscr{R}$ and $\mathscr{L}$ is a noncentral Lie ideal of $\mathscr{R}.$ If $$\bigr[\mathscr{F}(\mathcal{G}(u)u)-u\mathscr{H}(u),u\bigr]_n=0$$ for all $u \in \mathscr{L},$ then we describe the structure of the maps $\mathscr{F}, \mathcal{G},$ and $\mathscr{H}.$