Orthogonal (𝜎, 𝜏)-Derivations on Semiprime Γ –
Semirings

V.S.V. Krishna Murty; C. Jaya Subba Reddy; K. Chennakesavulu

doi:https://doi.org/10.14445/22315373/IJMTT-V70I7P103

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 70 | Issue 7 | Year 2024 | Article Id. IJMTT-V70I7P103 | DOI : https://doi.org/10.14445/22315373/IJMTT-V70I7P103

Orthogonal (𝜎, 𝜏)-Derivations on Semiprime Γ – Semirings

V.S.V. Krishna Murty, C. Jaya Subba Reddy, K. Chennakesavulu

Received	Revised	Accepted	Published
20 May 2024	27 Jun 2024	13 Jul 2024	31 Jul 2024

Abstract

Assume S is a semiprime Γ-semiring and d: S→S is an additive mapping that obeys 𝑑(𝑢𝛼𝑣) = 𝑑(𝑢)𝛼𝜎(𝑣)+ 𝜏(𝑢)𝛼𝑑(𝑣) for all u , 𝑣 ∈ 𝑆, 𝛼 ∈ 𝛤 , then d is termed as (𝜎, 𝜏)- derivation on S. This paper introduces orthogonal (𝜎, 𝜏)- derivations within semiprime Γ-semirings and provides several characterizations of these semirings. It also establishes the criteria under which two (𝜎, 𝜏)-derivations can be deemed orthogonal.

Keywords

(𝝈, 𝝉)-derivation, Orthogonal (𝝈, 𝝉)-derivation, 𝜞 –semiring.

References

[1] Abdulrahman H. Majeed, and Shahed Ali Hamil, “Orthogonal Generalized Derivations on Γ-Semirings,” Journal of Physics: Conference Series: Imam Al-Kadhum International Conference for Modern Applications of Information and Communication Technology (MAICT), Baghdad, Iraq, vol. 1530, 2020.
[CrossRef] [Google Scholar] [Publisher Link]
[2] B. Venkateswarlu, M. Murali Krishna Rao, and Y. Adi Narayana, “Orthogonal Derivations on Γ – Semirings,” Bulletin of the International Mathematical Virtual Institute, vol. 8, no. 3, pp. 543-552, 2018.
[Google Scholar]
[3] B. Venkateswarlu, M. Murali Krishna Rao, and Y. Adi Narayana, “Orthogonal Reverse Derivations on Semiprime Γ-Semirings,” Mathematical Sciences and Applications E-Notes, vol. 7, no. 1, pp. 71-77, 2019.
[CrossRef] [Google Scholar] [Publisher Link]
[4] C. Jaya Subba Reddy, and V.S.V. Krishna Murty, “Orthogonal Symmetric Reverse Bi-(𝜎, 𝜏)- Derivations in Semiprime Rings,” Tuijin Jishu/Journal of Propulsion Technology, vol. 45, no. 1, pp. 5133-5138, 2024.
[Google Scholar] [Publisher Link]
[5] C. Jaya Subba Reddy, and V.S.V. Krishna Murty, “Orthogonality of Generalized Reverse(𝜎, 𝜏)- Derivations in Semiprime 𝜞–Rings,” Journal of Nonlinear Analysis and Optimization, vol. 15, no. 4, pp. 20-28, 2024.
[Google Scholar] [Publisher Link]
[6] C. Jaya Subba Reddy, V.S.V. Krishna Murty, and K. Chennakesavulu, “Orthogonal Generalized Symmetric Reverse Bi-(𝜎, 𝜏)- Derivations in Semi Prime Rings,” Communications on Applied Nonlinear Analysis, vol. 31, no. 3s, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[7] C. Jaya Subba Reddy, and V.S.V. Krishna Murty, “Orthogonal generalized Symmetric Reverse Biderivations in Semiprime Rings,” Journal of Applied and Pure Mathematics, vol. 6, no. 3-4, pp. 155-165, 2024, in press.
[Publisher Link]
[8] Harry S. Vandiver, “Note on a Simple Type of Algebra in Which the Cancellation Law of Addition does not Hold,” Bulletin of the American Mathematical Society, vol. 40, no. 12, pp. 914-920, 1934.
[Google Scholar] [Publisher Link]
[9] Kalyan Kumar Dey, Akhil Chandra Paul, and Isamiddin S. Rakhimov, “Orthogonal Generalized Derivations in Semiprime Gamma Near-Rings,” International Journal of Algebra, vol. 6, no. 23, pp. 1127-1134, 2012.
[Google Scholar] [Publisher Link]
[10] M.K. Sen, “On Γ-Semigroups,” Proceeding of International Conference on Algebra and its Applications, New Delhi, pp.301- 308, 1981.
[Google Scholar]
[11] M.A. Javed, M. Aslam, and M. Hussain, “On Derivations of Prime Γ-Semirings,” Southeast Asian Bulletin of Mathematics, vol. 37, no. 6, 2013.
[Google Scholar]
[12] M. Murali Krishna Rao, “𝚪-Semirings-I,” Southeast Asian Bulletin of Mathematics, vol. 19, no. 1, pp. 49-54, 1995.
[Google Scholar] [Publisher Link]
[13] M. Murali Krishna Rao, “𝚪-Semirings-II,” Southeast Asian Bulletin of Mathematics, vol. 21, no. 3, pp. 281-287, 1997.
[Publisher Link]
[14] Nobuo Nobusawa, “On a Generalization of the Ring Theory,” Osaka Journal of Mathematics, vol. 1, no. 1, pp. 81-89, 1964.
[Google Scholar] [Publisher Link]
[15] N. Suganthameena, and M. Chandramouleeswaran, “Orthogonal Derivations on Semirings,” International Journal of Contemporary Mathematical Science, vol. 9, no. 13, pp. 645-651, 2014.
[CrossRef] [Google Scholar] [Publisher Link]
[16] Shuliang Huang, “On Orthogonal Generalized (σ, τ)-Derivations of Semiprime Near-Rings,” Kyungpook Mathematical Journal, vol. 50, no. 3, pp. 379-387, 2010.
[CrossRef] [Google Scholar] [Publisher Link]

Citation :

V.S.V. Krishna Murty, C. Jaya Subba Reddy, K. Chennakesavulu, "Orthogonal (𝜎, 𝜏)-Derivations on Semiprime Γ – Semirings," International Journal of Mathematics Trends and Technology (IJMTT), vol. 70, no. 7, pp. 13-17, 2024. Crossref, https://doi.org/10.14445/22315373/IJMTT-V70I7P103