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JORDAN GENERALIZED DERIVATIONS ON TRIVIAL EXTENSION ALGEBRAS

  • Bahmani, Mohammad Ali (Department of Pure Mathematics Ferdowsi University of Mashhad) ;
  • Bennis, Driss (Department of Mathematics Faculty of Sciences) ;
  • Vishki, Hamid Reza Ebrahimi (Department of Pure Mathematics Centre of Excellence in Analysis on Algebraic Structures (CEAAS) Ferdowsi University of Mashhad) ;
  • Attar, Azam Erfanian (Department of Pure Mathematics Ferdowsi University of Mashhad) ;
  • Fahid, Barahim (Department of Mathematics Faculty of Sciences B.P. 1014, Mohammed V University in Rabat)
  • Received : 2017.07.11
  • Accepted : 2017.11.02
  • Published : 2018.07.31

Abstract

In this paper, we investigate the problem of describing the form of Jordan generalized derivations on trivial extension algebras. One of the main results shows, under some conditions, that every Jordan generalized derivation on a trivial extension algebra is the sum of a generalized derivation and an antiderivation. This result extends the study of Jordan generalized derivations on triangular algebras (see [12]), and also it can be considered as a "generalized" counterpart of the results given on Jordan derivations of a trivial extension algebra (see [11]).

Keywords

trivial extension algebra;triangular algebra;Jordan derivation;Jordan generalized derivation;f-generalized derivation

References

  1. K. I. Beidar and Y. Fong, On additive isomorphisms of prime rings preserving polynomials, J. Algebra 217 (1999), no. 2, 650-667. https://doi.org/10.1006/jabr.1998.7833
  2. D. Benkovic, A note on f-derivations of triangular algebras, Aequationes Math. 89 (2015), no. 4, 1207-1211. https://doi.org/10.1007/s00010-014-0298-y
  3. D. Benkovic, Lie triple derivations of unital algebras with idempotents, Linear Multilinear Algebra 63 (2015), no. 1, 141-165. https://doi.org/10.1080/03081087.2013.851200
  4. D. Benkovic and D. Eremita, Commuting traces and commutativity preserving maps on triangular algebras, J. Algebra 280 (2004), no. 2, 797-824. https://doi.org/10.1016/j.jalgebra.2004.06.019
  5. D. Bennis and B. Fahid, Derivations and the first cohomology group of trivial extension algebras, Mediterr. J. Math. 14 (2017), no. 4, Art. 150, 18 pp. https://doi.org/10.1007/s00009-016-0805-6
  6. G. F. Birkenmeier, J. K. Park, and S. T. Rizvi, Extensions of Rings and Modules, Birkhauser/Springer, New York, 2013.
  7. W.-S. Cheung, Mappings on triangular algebras, ProQuest LLC, Ann Arbor, MI, 2000.
  8. W.-S. Cheung, Lie derivations of triangular algebras, Linear Multilinear Algebra 51 (2003), no. 3, 299-310. https://doi.org/10.1080/0308108031000096993
  9. H. R. Ebrahimi Vishki, M. Mirzavaziri, and F. Moafian, Jordan higher derivations on trivial extension algebras, Commun. Korean Math. Soc. 31 (2016), no. 2, 247-259. https://doi.org/10.4134/CKMS.2016.31.2.247
  10. M. Fosner and D. Ilisevic, On Jordan triple derivations and related mappings, Mediterr. J. Math. 5 (2008), no. 4, 415-427. https://doi.org/10.1007/s00009-008-0159-9
  11. H. Ghahramani, Jordan derivations on trivial extensions, Bull. Iranian Math. Soc. 39 (2013), no. 4, 635-645.
  12. Y. Li and D. Benkonic, Jordan generalized derivations on triangular algebras, Linear Multilinear Algebra 59 (2011), no. 8, 841-849. https://doi.org/10.1080/03081087.2010.507600
  13. H. Li and Y. Wang, Generalized Lie triple derivations, Linear Multilinear Algebra 59 (2011), no. 3, 237-247. https://doi.org/10.1080/03081080903350153
  14. A. H. Mokhtari, F. Moafian, and H. R. Ebrahimi Vishki, Lie derivations on trivial extension algebras, Ann. Math. Sil. 31 (2017), no. 1, 141-153.
  15. A. Nakajima, On generalized higher derivations, Turkish J. Math. 24 (2000), no. 3, 295-311.
  16. J. Wu and S. Lu, Generalized Jordan derivations on prime rings and standard operator algebras, Taiwanese J. Math. 7 (2003), no. 4, 605-613. https://doi.org/10.11650/twjm/1500407580
  17. Y. Zhang, Weak amenability of module extensions of Banach algebras, Trans. Amer. Math. Soc. 354 (2002), no. 10, 4131-4151. https://doi.org/10.1090/S0002-9947-02-03039-8