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

Authors : Shakir ALİ, Mohammad Salahuddin KHAN

Pages : 23-39

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Publication Date : 2013-06-01

Article Type : Research Paper

Abstract :Let M be a Γ-ring and σ, τ be endomorphisms of M. An additive mapping d : M −→ M is called a insert ignore into journalissuearticles values(σ, τ);-derivation if dinsert ignore into journalissuearticles values(xαy); = dinsert ignore into journalissuearticles values(x);ασinsert ignore into journalissuearticles values(y); + τinsert ignore into journalissuearticles values(x);αdinsert ignore into journalissuearticles values(y); holds for all x, y ∈ M and α ∈ Γ. An additive mapping F : M −→ M is called a generalized insert ignore into journalissuearticles values(σ, τ);-derivation if there exists a insert ignore into journalissuearticles values(σ, τ);- derivation d : M −→ M such that Finsert ignore into journalissuearticles values(xαy); = Finsert ignore into journalissuearticles values(x);ασinsert ignore into journalissuearticles values(y); + τinsert ignore into journalissuearticles values(x);αdinsert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(σ, τ);-derivations and orthogonal generalized insert ignore into journalissuearticles values(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.
Keywords : semiprime Γ ring, derivation, orthogonal derivation, orthogonal σ, τ, derivation, orthogonal generalized derivation, orthogonal generalized σ, τ, derivation