On derivations over trivial extensions





Derivation, trivial extension, prime ring


In this paper, we investigate the structure of derivations over trivial extensions. We provide a detailed analysis of the structure of derivations on trivial extensions, the centre of trivial extensions, and the conditions for a trivial extension to be prime. Additionally, we examine the structure of derivations on trivial extensions when the underlying ring, $R$, is a prime ring, under the conditions of Herstein's Theorem, Posner's Theorem, and Bell's theorem.

Author Biography

Brahim Boudine, Sidi Mohamed Ben Abdellah University.

Faculty of sciences Dhar El Mahraz


bibitem{ADA} S. Ali, N. A. Dar, and M. Asci, textit{On derivations and commutativity of prime rings with involution}, Georgian Math. J. textbf{23} (2016), no. 1, 9--14.

bibitem{Anderson} D. D. Anderson, and M. Winders, textit{Idealization of a module}, Journal of commutative algebra textbf{1} (2009), no. 1, 3--56.

bibitem{bennis2} M. A. Bahmani, D. Bennis, H. R. E. Vishki, A. E. Attar, and B. Fahid, textit{Jordan generalized derivations on trivial extension algebras}, Communications of the Korean Mathematical Society textbf{33} (2018), no. 3, 721--739.

bibitem{bell} H. E. Bell, and G. Mason, textit{derivations in near-rings}, North-Holland Mathematics Studies textbf{137} (1987) 31--35.

bibitem{bennis1} D. Bennis, H. R. E. Vishki, B. Fahid, A. A. Khadem-Maboudi, and A. H. Mokhtari, textit{Lie generalized derivations on trivial extension algebras}, Bollettino dell'Unione Matematica Italiana textbf{12} (2019) 441--452.

bibitem{Bresar} M. Bresar, and J. Vukman, textit{Jordan $(theta,phi)$-derivations}, Glas. Mat. textbf{26} (1991), no. 46, 13--17.

bibitem{herstein} I. N. Herstein, textit{A note on derivations II}, Can. Math. Bull. textbf{22} (1979), no. 4, 509--511.

bibitem{Lee} P. H. Lee, and T. K. Lee, textit{On derivations of prime rings}, Chinese J. Math. textbf{9} (1981), no. 2, 107--110.

bibitem{MOZ1} A. Mamouni, L. Oukhtite, M. Zerra, textit{On derivations involving prime ideals and commutativity in rings}, Sao Paulo Journal of Mathematical Sciences textbf{14} (2020), no. 2, 675--688.

bibitem{MOZ2} A. Mamouni, L. Oukhtite, M. Zerra, textit{Certain algebraic identities on prime rings with involution}, Communications in Algebra textbf{49} (2021), no. 7, 2976--2986.

bibitem{oukhtite} M. A. Idrissi, L. Oukhtite, textit{Derivations over amalgamated algebras along an ideal}, Communications in Algebra textbf{48} (2020), no. 3, 1224--1230. DOI: https://doi.org/10.1080/00927872.2019.1677696

bibitem{posner} E. C. Posner, textit{Derivations in prime rings}, Proc. Amer.Math. Soc. textbf{8} (1957), no. 6, 1093--1100.



How to Cite

B. Boudine and M. Zerra, “On derivations over trivial extensions”, Proyecciones (Antofagasta, On line), vol. 43, no. 2, pp. 459-472, Apr. 2024.




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