Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings

Abstract

Let R be a noncommutative prime ring, I a nonzero left ideal of R, L a non-central Lie ideal of R, U the left Utumi quotient ring of R and the extended centroid of R. Let G be a generalized derivation of R and g be a derivation of R and fixed positive integers. In the present paper, we describe the structure of R and all possible forms of G and g in the following cases: for all and for all