Product of generalized derivations with Engel condition on Lie ideals in prime rings

Abstract

This paper represents the structures of generalized derivations satisfying an identity with [Formula: see text]-Engel condition. To prove our result, we take [Formula: see text] as a prime ring with [Formula: see text], and [Formula: see text] as Utumi quotient ring of [Formula: see text]. The center of [Formula: see text] is [Formula: see text] which is called extended centroid of [Formula: see text]. Let [Formula: see text] be a non-central Lie ideal of [Formula: see text]. Suppose that [Formula: see text] and [Formula: see text] are two nonzero generalized derivations of [Formula: see text] and [Formula: see text], any fixed integer. If [Formula: see text] for all [Formula: see text], then one of the below-mentioned conclusions holds: (1) [Formula: see text] and [Formula: see text] for all [Formula: see text] and for some [Formula: see text] with [Formula: see text]; (2) [Formula: see text] satisfies [Formula: see text].