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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.41 no.3 Antofagasta June 2022

http://dx.doi.org/10.22199/issn.0717-6279-4399 

Artículos

Structure of a quotient ring R/P and its relation with generalized derivations of R

Karim Bouchannafa1 

Abdellah Mamouni2 

Lahcen Oukhtite3 

1 Department of Mathematics, Faculty of Science and Technology, Sidi Mohamed Ben Abdellah University, Box 2202, Fez, Morocco. E-mail: bouchannafa.k@gmail.com

2 Department of Mathematics, Faculty of Science, University Moulay Ismaïl, Meknes, Morocco. email: a.mamouni.fste@gmail.com

3 Department of Mathematics, Faculty of Science and Technology, Sidi Mohamed Ben Abdellah University, Box 2202, Fez, Morocco

Abstract

The fundamental aim of this paper is to investigate the structure of a quotient ring R/P where R is an arbitrary ring and P is a prime ideal of R. More precisely, we will characterize the commutativity of R/P via the behavior of generalized derivations of R satisfying algebraic identities involving the prime ideal P. Moreover, various wellknown results characterizing the commutativity of prime (semi-prime)rings have been extended. Furthermore, examples are given to prove that the restrictions imposed on the hypothesis of the various theorems were not superfluous.

Keywords: Quotient ring; prime ideal; generalized derivations; commutativity

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References

[1] F. A. A. Almahdi, A. Mamouni and M. Tamekkante, “A Generalization of Posner’s Theorem on Derivations in Rings”, Indian Journal of Pure and Applied Mathematics, vol. 51, no. 1, pp. 187-194, 2020. [ Links ]

[2] M. Ashraf and A. Khan, “Commutativity of ∗-prime rings with generalized derivations”, Rendiconti del Seminario Matematico della Università di Padova, vol. 125, pp. 71-79, 2011. [ Links ]

[3] M. Ashraf, N. Rehman, A. Shakir and M. Rahman Mozumder, “On semiprime rings with generalized derivations”, Boletim da Sociedade Paranaense de Matemática, vol. 28, no. 2, pp. 25-32, 2010. [ Links ]

[4] M. Ashraf , A. Ali and S. Ali, “Some commutativity theorems for rings with generalized derivations”, Southeast Asian Bulletin of Mathematics, vol. 31, no. 3, pp. 415-421, 2007. [ Links ]

[5] M. Ashraf andN. Rehman , “On commutativity of rings with derivations”, Results in Mathematics, vol. 42, no. 1-2, pp. 3-8, 2002. [ Links ]

[6] H. E. Bell and W. S. Martindale III, “Centralizing mappings of semiprime rings. Canadian Mathematical Bulletin, vol. 30, no. 1, pp. 92-101, 1987. [ Links ]

[7] M. Brešar, “On the distance of the composition of two derivations to the generalized derivations”, Glasgow Mathematical Journal, vol. 33, no. 1, pp. 89-93, 1991. [ Links ]

[8] M. Brešar, “Centralizing mappings and derivations in prime rings”, Jornal of algebra, vol. 156, no. 2, pp. 385-394, 1993. [ Links ]

[9] I. N. Herstein, “A note on derivations”, Canadian Mathematical Bulletin , vol. 21, no. 3, pp. 369-370, 1978. [ Links ]

[10] C. Lanski, “Differential identities, Lie ideals and Posner’s theorems”, Pacific Journal of Mathematics, vol. 134, no. 2, pp. 275-297, 1988. [ Links ]

[11] J. Mayne, Centralizing automorphisms of prime rings. Canadian Mathematical Bulletin , vol. 19, no. 1, pp. 113-115, 1976. [ Links ]

[12] A. Mamouni, L. Oukhtite and B. Nejjar, “On ∗-semiderivations and ∗-generalized semiderivations”, Journal of Algebra and its Applications, vol. 16, no. 4, Art. ID. 1750075, 2017 [ Links ]

[13] L. Oukhtite, “Posners second theorem for Jordan ideals in rings with involution”, Expositiones Mathematicae, vol. 29, no. 4, pp. 415-419, 2011. [ Links ]

[14] E. C. Posner, “Derivations in prime rings”, Proceedings of the American Mathematical Society, vol. 8, pp. 1093-1100, 1957. [ Links ]

[15] M. A. Quadri, M. S. Khan andN. Rehman , “Generalized derivations and commutativity of prime rings”, Indian Journal of Pure and Applied Mathematics , vol. 34, no. 9, pp. 1393-1396, 2003. [ Links ]

[16] N. Ur-Rehman, “On generalized derivations as homomorphisms and anti-homomorphisms”, Glasnik Matematički, vol. 39, no. 59, pp. 27-30, 2004. [ Links ]

[17] S. K. Tiwari, R. K. Sharma and B. Dhara, “Identities related to gener-alized derivation on ideal in prime rings”, Beiträge zur Algebra und Geometrie, vol. 57, no. 4, pp. 809-821, 2016. [ Links ]

Received: August 30, 2020; Accepted: December 30, 2021

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