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

Year 2013, Volume: 13 Issue: 13, 23 - 39, 01.06.2013

Shakir Ali Mohammad Salahuddin Khan

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.

Keywords

semiprime Γ-ring, derivation, orthogonal derivation, orthogonal (σ, τ)-derivation, orthogonal generalized derivation, orthogonal generalized (σ, τ)-derivation

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• Shakir Ali, Mohammad Salahuddin Khan Department of Mathematics Aligarh Muslim University Aligarh-202002, India e-mails: [e-mail] (Shakir Ali) [e-mail] (Mohammad Salahuddin Khan)