Abstract
Let R be a prime ring of characteristic different from two with center Z(R). In the present paper we study the case when a generalized derivation F associated with a nonzero derivation d of R satisfies [F([x, y](k)), [x, y](k)] is an element of Z(R) for all x, y in some appropriate subset of R, where k is a fixed positive integer. We obtain a description of the structure of R and information on the form of F in terms of the standard identity s(4) and the multiplication by the specific element from the extended centroid.