On Generalized (σ,τ)‎-Derivations in Semiprime Rings

Abstract EN

Let R be a semiprime ring, I a nonzero ideal of R, and σ, τ two epimorphisms of R.

An additive mapping F:R→R is generalized (σ,τ)-derivation on R if there exists a (σ,τ)-derivation d:R→R such that F(xy)=F(x)σ(y)+τ(x)d(y) holds for all x,y∈R.

In this paper, it is shown that if τ(I)d(I)≠0, then R contains a nonzero central ideal of R, if one of the following holds: (i) F[x,y]=±(x∘y)σ,τ; (ii) F(x∘y)=±[x,y]σ,τ; (iii) F[x,y]=±[F(x),y]σ,τ; (iv) F(x∘y)=±(F(x)∘y)σ,τ; (v) F[x,y]=±[σ(y),G(x)] for all x,y∈I.