An Engel condition with b-generalized derivations for Lie ideals

Abstract

Let [Formula: see text] be a prime ring with the extended centroid [Formula: see text], [Formula: see text] a noncommutative Lie ideal of [Formula: see text] and [Formula: see text] a nonzero [Formula: see text]-generalized derivation of [Formula: see text]. For [Formula: see text], let [Formula: see text]. We prove that if [Formula: see text] for all [Formula: see text], where [Formula: see text] are fixed positive integers, then there exists [Formula: see text] such that [Formula: see text] for all [Formula: see text] except when [Formula: see text], the [Formula: see text] matrix ring over a field [Formula: see text]. The analogous result for generalized skew derivations is also described. Our theorems naturally generalize the cases of derivations and skew derivations obtained by Lanski in [C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 75–80, Skew derivations and Engel conditions, Comm. Algebra 42 (2014), 139–152.]