On one sided ideals of a semiprime ring with generalized derivations

Let R be a ring with center Z(R). An additive mapping F: R → R is said to be a generalized derivation on R if there exists a derivation d: R → R such that F(xy) = F(x)y + xd(y), for all x, y ∈ R (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) ≠ (0) then R contains some nonzero central ideal, if one of the following conditions holds: (1) R is 2-torsion free and F(xy) ∈ Z(R), for all x,y ∈ U, unless F(U)U = UF(U) = Ud(U) = (0); (2) F(xy) ∓ yx ∈ Z(R), for all x,y ∈ U; (3) F(xy) ∓ [x,y] ∈ Z(R), for all x,y ∈ U; (4) F ≠ 0 and F([x,y]) = 0, for all x, y ∈ U, unless Ud(U) = (0); (5) F ≠ 0 and F([x, y]) ∈ Z(R), for all x, y ∈ U, unless either d(Z(R))U = (0) or Ud(U) = (0)n. © 2012 Springer Basel AG.