Generalized derivations on Lie ideals with annihilating Engel conditions

Document Type : Original Article


1 School of Liberal Studies, Ambedkar University Delhi, Delhi-110006, India

2 Department of Mathematics, Ambedkar University Delhi, Delhi-110006, India



Let $\mathcal{R}$ be a non-commutative prime ring with characteristic different from $2$, $\mathcal{U}$ be the Utumi quotient ring of $\mathcal{R}$ and $\mathcal{C}$ be the extended centroid of $\mathcal{R}$. Let $\mathcal{G}$ be a generalized derivation on $\mathcal{R}$, $\mathcal{L}$ be a non-central Lie ideal of $\mathcal{R}$, $0 \neq c \in\mathcal{R}$ and $ n, r, s,t$ are fixed positive integers. If $c u^s[\mathcal{G}(u^n),u^r]_ku^t =0$, for all $u \in\mathcal{L}$, then one of the following holds:
(1) $\mathcal{R}$ satisfies $s_4$.
(2) There exists $\lambda \in \mathcal{C}$ s.t. $\mathcal{G}(\zeta)= \lambda \zeta$ for all $\zeta \in \mathcal{R}$.
(3) If $\mathcal{C}$ is finite field, $\mathcal{R} \cong M_{l}(\mathcal{C})$, a $l \times l$ matrix ring over $\mathcal{C}$ for $l>2$.