Remarks on Generalized Derivations in Prime and Semiprime Rings

Abstract

Let R be a ring with center Z and I a nonzero ideal of R. An additive mapping F:R→R is called a generalized derivation of R if there exists a derivation d:R→R such that F(xy)=F(x)y+xd(y) for all x,y∈R. In the present paper, we prove that if F([x,y])=±[x,y] for all x,y∈I or F(x∘y)=±(x∘y) for all x,y∈I, then the semiprime ring R must contains a nonzero central ideal, provided d(I)≠0. In case R is prime ring, R must be commutative, provided d≠0. The cases (i) F([x,y])±[x,y]∈Z and (ii) F(x∘y)±(x∘y)∈Z for all x,y∈I are also studied.