On Semiprime Rings with Generalized Derivations

Let R be a ring and F and G be generalized derivations of R with associated derivations d and g respectively. In the present paper, we shall investigate the commutativity of semiprime ring R admitting generalized derivations F and G satisfying any one of the properties: (i)F(x)x = xG(x); (ii) [F(x); d(y)] = [x; y]; (iii) F([x; y]) = [d(x); F(y)]; (iv) d(x)F(y) = xy; (v) F(x2) = x2; (vi) [F(x); y] = [x;G(y)]; (vii) F([x; y]) = [F(x); y]+[d(y); x] and (viii) F(x ◦ y) = F(x) ◦ y − d(y) ◦ x for all x; y in some appropriate subset of R.