ON LIE IDEALS AND GENERALIZED DERIVATIONS OF PRIME RINGS

Abstract

Let R be a ring and S a nonempty subset of R. An additive mapping F : R → R is called a generalized derivation on S if there exists a derivation d : R → R such that F(xy)=F(x)y + xd(y), for all x,y ∈ S. Suppose that U is a Lie ideal of R with the property that u2 ∈ U, for all u ∈ U. In the present paper, we prove that if R is a prime ring with characteristic different from 2 admitting a generalized derivation F satisfy any one of the properties: (i) F(uv)−uv ∈ Z(R), (ii) F(uv)+uv ∈ Z(R), (iii) F(uv) − vu ∈ Z(R) and ( iv) F(uv)+vu ∈ Z(R), for allu,v ∈ U, thenU must be central