ON ORTHOGONAL (σ, τ )-DERIVATIONS IN SEMIPRIME Γ-RINGS

Abstract

Let M be a Γ-ring and σ, τ be endomorphisms of M. An additive mapping d : M −→ M is called a (σ, τ)-derivation if d(xαy) = d(x)ασ(y) + τ(x)αd(y) holds for all x, y ∈ M and α ∈ Γ. An additive mapping F : M −→ M is called a generalized (σ, τ)-derivation if there exists a (σ, τ)- derivation d : M −→ M such that F(xαy) = F(x)ασ(y) + τ(x)αd(y) holds for all x, y ∈ M and α ∈ Γ. In this paper, some known results on orthogonal derivations and orthogonal generalized derivations of semiprime Γ-rings are extended to orthogonal (σ, τ)-derivations and orthogonal generalized (σ, τ)- derivations. Moreover, we present some examples which demonstrate that the restrictions imposed on the hypotheses of some of our results are not superfluous.