Generalized Derivations of Prime Rings

Let R be an associative prime ring, U a Lie ideal such that u2∈U for all u∈U. An additive function F:R→R is called a generalized derivation if there exists a derivation d:R→R such that F(xy)=F(x)y+xd(y) holds for all x,y∈R. In this paper, we prove that d=0 or U⊆Z(R) if any one of the following conditions holds: (1) d(x)∘F(y)=0, (2) [d(x),F(y)=0], (3) either d(x)∘F(y)=x∘y or d(x)∘F(y)+x∘y=0, (4) either d(x)∘F(y)=[x,y] or d(x)∘F(y)+[x,y]=0, (5) either d(x)∘F(y)−xy∈Z(R) or d(x)∘F(y)+xy∈Z(R), (6) either [d(x),F(y)]=[x,y] or [d(x),F(y)]+[x,y]=0, (7) either [d(x),F(y)]=x∘y or [d(x),F(y)]+x∘y=0 for all x,y∈U