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

• Bouchannafa, Karim ^[1] ; Mamouni, Abdellah ^[2] ; Oukhtite, Lahcen ^[1]
  1. [1] Sidi Mohamed Ben Abdellah University

     Sidi Mohamed Ben Abdellah University

     Fes-Medina, Marruecos

     
  2. [2] Moulay Ismaïl University.
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 41, Nº. 3, 2022, págs. 623-642
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-4399
• Enlaces
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• Resumen

  • 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.

• Referencias bibliográficas
  • F. A. A. Almahdi, A. Mamouni and M. Tamekkante, “A Generalization of Posner’s Theorem on Derivations in Rings”, Indian Journal of Pure and...
  • M. Ashraf and A. Khan, “Commutativity of ∗-prime rings with generalized derivations”, Rendiconti del Seminario Matematico della Università...
  • M. Ashraf, N. Rehman, A. Shakir and M. Rahman Mozumder, “On semiprime rings with generalized derivations”, Boletim da Sociedade Paranaense...
  • M. Ashraf, A. Ali and S. Ali, “Some commutativity theorems for rings with generalized derivations”, Southeast Asian Bulletin of Mathematics,...
  • M. Ashraf and N. Rehman, “On commutativity of rings with derivations”, Results in Mathematics, vol. 42, no. 1-2, pp. 3-8, 2002.
  • H. E. Bell and W. S. Martindale III, “Centralizing mappings of semiprime rings. Canadian Mathematical Bulletin, vol. 30, no. 1, pp. 92-101,...
  • M. Brešar, “On the distance of the composition of two derivations to the generalized derivations”, Glasgow Mathematical Journal, vol. 33,...
  • M. Brešar, “Centralizing mappings and derivations in prime rings”, Jornal of algebra, vol. 156, no. 2, pp. 385-394, 1993.
  • I. N. Herstein, “A note on derivations”, Canadian Mathematical Bulletin, vol. 21, no. 3, pp. 369-370, 1978.
  • C. Lanski, “Differential identities, Lie ideals and Posner’s theorems”, Pacific Journal of Mathematics, vol. 134, no. 2, pp. 275-297, 1988.
  • J. Mayne, Centralizing automorphisms of prime rings. Canadian Mathematical Bulletin, vol. 19, no. 1, pp. 113-115, 1976.
  • A. Mamouni, L. Oukhtite and B. Nejjar, “On ∗-semiderivations and ∗-generalized semiderivations”, Journal of Algebra and its Applications,...
  • L. Oukhtite, “Posners second theorem for Jordan ideals in rings with involution”, Expositiones Mathematicae, vol. 29, Nno. 4, pp. 415-419,...
  • E. C. Posner, “Derivations in prime rings”, Proceedings of the American Mathematical Society, vol. 8, pp. 1093-1100, 1957.
  • M. A. Quadri, M. S. Khan and N. Rehman, “Generalized derivations and commutativity of prime rings”, Indian Journal of Pure and Applied Mathematics,...
  • N. Ur-Rehman, “On generalized derivations as homomorphisms and anti-homomorphisms”, Glasnik Matematički, vol. 39, no. 59, pp. 27-30, 2004.
  • S. K. Tiwari, R. K. Sharma and B. Dhara, “Identities related to gener-alized derivation on ideal in prime rings”, Beiträge zur Algebra und...