Superderivations and Jordan superderivations of generalized quaternion algebras

Document Type : Special Issue: First Joint IIIMT-Algebra Forum Conference 2023

Author

Department of Mathematics and Statistics, Shoushtar Branch, Islamic Azad University, Shoushtar, Iran.

Abstract

Let $H_{\alpha,\beta}$ be the generalized quaternion algebra over a unitary commutative ring. This paper aims to consider superderivations and Jordan superderivations of $H_{\alpha,\beta}$ and hence to obtain the superalgebra $Der_{s} (H_{\alpha,\beta})$ of superderivations and $Der_{Js} (H_{\alpha,\beta})$ of Jordan superderivations of $H_{\alpha,\beta}$. It turns out that on generalized quaternion algebras, any superderivation is inner.  In particular, there exist Jordan superderivations that are not superderivations.

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References

[1] Bresar, M. (1988). Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104, 1003{1006.
[2] Cheraghpour, H. and Ghosseiri, M.N. (2019). On superderivations and super-biderivations of trivial extensions and triangular matrix rings, Comm. Algebra. 47(4), 1662{1670. https://doi.org/10.1080/00927872.2018.1513016
[3] Fosner, A. and Fosner, M. (2009). 2-local superderivations on a superalgebra Mn(C), Monatsh. Math. 156, 307{311. https://doi.org/10.1007/s00605-008-0070-2
[4] Fosner, M. (2003). Jordan superderivations, Comm. Algebra. 31(9), 4533{4545. https://doi.org/10.1081/AGB-120022808
[5] Ghahramani, H., Ghosseiri, M.N. and Safari, S. (2017). Some questions concerning superderivations on Z2-graded rings, Aequationes Math. 91, 725{738. https://doi.org/10.1007/s00010-017-0480-0
[6] Ghahramani, H., Ghosseiri, M.N. and Heidari Zadeh, L. (2021). Generalized Derivations and Generalized Jordan Derivations of Quaternion Rings. Iran J Sci Technol Trans Sci. 45, 305{308. https://doi.org/10.1007/s40995-020-01046-4
[7] Ghahramani, H., Ghosseiri, M.N. and Safari, S. (2021). On derivations, biderivations and superbiderivations of quaternion rings, Rend. Circ. Mat. Palermo, II. Ser 70, 765{674. https://doi.org/10.1007/s12215-020-00518-1
[8] Herstein, I.N. (1957). Jordan derivations of prime rings. Proc. Amer. Math. Soc. 8, 1104{1110.
[9] Kizil, E. and Alagoz, Y. (2019). Derivations of generalized quaternion algebra. Turk. J. Math. 43(5), 2649{2657. https://doi.org/10.3906/mat-1905-86
[10] Lam,T.Y. (2005). Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, American Mathematical Society. https://doi.org/10.1090/gsm/067
[11] Li, F. and Wan, L. (2016). On P-Derivations and P-Jordan derivations of a ring, Ital. J. Pure Appl. Math. 36, 639{650.
[12] Zhang, J.H. and Yu, W.Y. (2006). Jordan derivations of triangular algebras, Linear Algebra Appl. 419, 251{255. https://doi.org/10.1016/j.laa.2006.04.015
Journal of Mahani Mathematical Research
Volume 13, Issue 5 - Serial Number 30
Special Issue: First Joint IIIMT-Algebra Forum Conference 2023
December 2024
Pages 1-9

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History

  • Receive Date: 30 April 2024
  • Revise Date: 22 July 2024
  • Accept Date: 04 August 2024

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