On commutativity of rings with generalized derivations

Let R be a prime ring, extended centroid C, Utumi quotient ring U, and m, n ≥ 1 are fixed positive integers, F a generalized derivation associated with a nonzero derivation d of R  . We study the case when one of the following holds: (i) F(x)∘md(y)=n(x∘y)F(x)∘md(y)=(x∘y)n and (ii) m(F(x)∘d(y))=n(x∘y)(F(x)∘d(y))m=(x∘y)n, for all x, y in some appropriate subset of R. We also examine the case where R is a semiprime ring.