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 (x(2)) = x(2), (vi) [F(x), y] = [x, G(y)], (vii) F([x, y]) = [F(x), y] [d(y), x] and (viii) F(x o y) = F(x) o y d(y) o x for all x, y in some appropriate subset of R.