Abstract
Let R be a non-commutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, L a non-central Lie ideal of R, G a non-zero generalized derivation of R.
If [G(u), u](n) = [G(u), u], for all u is an element of L, with n > 1, then one of the following holds: (1) R satisfies the standard identity s(4)(x(1),...,x(4)) and there exist alpha is an element of U and alpha is an element of C such that G(x) = ax + xa + ax for all x is an element of R;
(2) there exists gamma is an element of C such that G(x) = gamma x for all x is an element of R. (C) 2015 Mathematical Institute Slovak Academy of Sciences