Abstract
Let R be a ring with center Z(R). An additive mapping is said to be a generalized derivation on R if there exists a derivation such that F(xy) = F(x)y + xd(y), for all (the map d is called the derivation associated with F). Let R be a semiprime ring and U be a nonzero left ideal of R. In the present note we prove that if R admits a generalized derivation F, d is the derivation associated with F such that d(U) not equal (0) then R contains some nonzero central ideal, if one of the following conditions holds: (1) R is 2-torsion free and , for all , unless F(U)U = UF(U) = Ud(U) = (0); (2) , for all ; (3) , for all ; (4) F not equal 0 and F([x,y]) = 0, for all , unless Ud(U) = (0); (5) F not equal 0 and , for all , unless either d(Z(R))U = (0) or Ud(U) = (0)n.