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## Proyecciones (Antofagasta)

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*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*

^{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

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