ON ORTHOGONAL (σ, τ )-DERIVATIONS IN SEMIPRIME Γ-RINGS
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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- N. Arga¸c, A. Kaya and A. Kisir, (σ, τ )-derivations in prime rings, Math. J. Okayama Univ., 29 (1987), 173–177.
- N. Arga¸c, A. Nakajima and E. Alba¸s, On orthogonal generalized derivations of semiprime rings, Turkish J. Math., 28(2) (2004), 185–194.
- M. Ashraf and M. R. Jamal, Orthogonal derivations in Γ-rings, Advances in Algebra, 3(1) (2010), 1–6.
- M. Ashraf and M. R. Jamal, Orthogonal Generalized derivations in Γ-rings, Aligarh Bull. Math., 29(1) (2010), 41–46.
- M. Ashraf, A. Ali and Shakir Ali, On Lie ideals and generalized (θ, φ)- derivations in prime rings, Comm. Algebra, 32(8) (2004), 2977–2985.
- W. E. Barnes, On the Γ-rings of Nobusawa, Pacific J. Math., 18(3) (1966), –422.
- M. Breˇsar and J. Vukman, Orthogonal derivation and an extension of a theorem of Posner, Rad. Mat., 5(2) (1989), 237–246.
- Y. C¸ even and M. A. ¨Ozt¨urk, On Jordan generalized derivation in gamma rings, Hacet. J. Math. Stat., 33 (2004), 11–14.
- F. J. Jing, On derivations of Γ-rings, Qufu Shifan Daxue Xuebao Ziran Kexue Ban, 13(4) (1987), 159–161.
- N. Nobusawa, On a generalization of the ring theory, Osaka J. Math., 1 (1964), –89.
- M. A. ¨Ozt¨urk, M. Sapanci, M. Soyt¨urk and K. H. Kim, Symmetric bi-derivation on prime Gamma-rings, Sci. Math., 3(2) (2000), 273–281.
- M. S. Yenig¨ul and N. Arga¸c, On ideals and orthogonal derivations, J. of South- west China Normal Univ., 20 (1995), 137–140.
- Shakir Ali, Mohammad Salahuddin Khan Department of Mathematics Aligarh Muslim University Aligarh-202002, India e-mails: shakir.ali.mm@amu.ac.in (Shakir Ali) salahuddinkhan50@gmail.com (Mohammad Salahuddin Khan)
- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI