On generalized derivations as homomorphisms and anti-homomorphisms

The concept of derivations as well as generalized derivations (i.e. I a,b (x) = ax + xb, for all a, b R) have been generalized as an additive function F : R - R satisfying F (xy) = F (x)y + xd(y) for all x, y R, where d is a nonzero derivation on R. Such a function F is said to be a generalized derivation. In the present paper it is shown that: if R is 2-torsion free prime ring, I = 0 an ideal of R and F a generalized derivation of R such that either F (xy) = F (x)F (y) or F (xy) = F (y)F (x) for all x, y I, then R is commutative.