On one sided ideals of a semiprime ring with generalized derivations

Abstract

Let R be a ring with center Z(R). An additive mapping ${F : R \longrightarrow R}$ is said to be a generalized derivation on R if there exists a derivation ${d : R \longrightarrow R}$ such that F(xy) = F(x)y + xd(y), for all ${x, y \in R}$ (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 ${F(xy) \in Z(R)}$, for all ${x, y \in U}$, unless F(U)U = UF(U) = Ud(U) = (0); (2) ${F(xy) \mp yx \in Z(R)}$, for all ${x,y \in U}$; (3) ${F(xy) \mp [x,y] \in Z(R)}$, for all ${x,y \in U}$; (4) F ≠ 0 and F([x,y]) = 0, for all ${x, y \in U}$, unless Ud(U) = (0); (5) F ≠ 0 and ${F([x, y]) \in Z(R)}$, for all ${x, y \in U}$, unless either d(Z(R))U = (0) or Ud(U) = (0)n.