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

Let R be a prime ring with its Utumi ring of quotients U, F a nonzero generalized derivation of R and L a noncentral Lie ideal of R. Suppose that [F(un1),un2,un3,…,unk]=0 for all u∈ L, where n1, n2, … , nk≥ 1 are fixed integers. Then one of the following holds:(1)there exists α∈ C such that F(x) = αx for all x∈ R;(2)R satisfies s4, the standard identity in four variables. Also we study the situation, when x∈ [ I, I] , where I is a nonzero left ideal of R. As an application we obtain some range inclusion results of continuous generalized derivations on Banach algebras. © 2016, African Mathematical Union and Springer-Verlag Berlin Heidelberg.