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

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 C=Z(U) the extended centroid of R. Let G be a generalized derivation of R and g be a derivation of R and m,n,r,k 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: [G(xm)xn+xng(xm), xr]k=0 for all XϵI; 2. char(R)≠2 and [G(xm)xn+xng(xm), xr]k=0 for all XϵI. © 2019, © 2019 Taylor & Francis Group, LLC.