Skip to main content
Log in

Lie (Jordan) derivations of arbitrary triangular algebras

  • Published:
Aequationes mathematicae Aims and scope Submit manuscript

Abstract

In this paper we construct a triangular algebra from a given triangular algebra, using the notion of maximal left (right) ring of quotients. As an application we give a description of Lie (Jordan) derivations of arbitrary triangular algebras through the constructed triangular algebra.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Beidar, K.I., Chebotar, M.A.: On Lie derivations of Lie ideals of prime rings. Isr. J. Math. 123, 131–148 (2001)

    Article  Google Scholar 

  2. Beidar, K.I., Martindale 3rd, W.S., Mikhale, A.: Rings with Generalized Identities. Marcel Dekker, NewYork (1996)

    Google Scholar 

  3. Benkovič, D.: Lie triple derivations of unital algebras with idempotents. Linear Multilinear Algebra 63, 141–165 (2004)

    Article  MathSciNet  Google Scholar 

  4. Benkovič, D.: Lie derivations on triangular matrices. Linear Multilinear Algebra 55, 619–629 (2007)

    Article  MathSciNet  Google Scholar 

  5. Benkovič, D.: Generalized Lie derivations on triangular rings. Linear Algebra Appl. 434, 1532–1544 (2011)

    Article  MathSciNet  Google Scholar 

  6. Benkovič, D.: A note on \(f\)-derivations of triangular algebras. Aequ. Math. 89, 1207–1211 (2015)

    Article  MathSciNet  Google Scholar 

  7. Benkovič, D., Eremita, D.: Commuting traces and commutativity preserving maps on triangular algebras. J. Algebra 280, 797–824 (2004)

    Article  MathSciNet  Google Scholar 

  8. Benkovič, D., Eremita, D.: Multiplicative Lie \(n\)-derivations of triangular rings. Linear Algebra Appl. 436, 4223–4240 (2012)

    Article  MathSciNet  Google Scholar 

  9. Brešar, M.: Commuting traces of biadditive mappings, commutativity preserving mappings and Lie mappings. Trans. Am. Math. Soc. 335, 525–546 (1993)

    Article  MathSciNet  Google Scholar 

  10. Brešar, M., Chebotar, M.A.: Martindale 3rd W. S.: Functional Identities. Birkhäuser Verlag, Basel (2007)

    Google Scholar 

  11. Brešar, M., Šemel, P.: Commuting traces of biadditive maps revisited. Commun. Algebra 31, 381–388 (2003)

    Article  MathSciNet  Google Scholar 

  12. Cheung, W.S.: Commuting maps of triangular algebras. J. Lond. Math. Soc. 63, 117–127 (2001)

    Article  MathSciNet  Google Scholar 

  13. Cheung, W.S.: Lie derivations of triangular algebras. Linear Multilinear Algebra 51, 299–310 (2003)

    Article  MathSciNet  Google Scholar 

  14. Eremita, D.: Functional identities of degree \(2\) in triangular rings. Linear Algebra Appl. 438, 584–597 (2013)

    Article  MathSciNet  Google Scholar 

  15. Eremita, D.: Functional identities of degree \(2\) in triangular rings revisited. Linear Multilinear Algebra 63, 534–553 (2015)

    Article  MathSciNet  Google Scholar 

  16. Eremita, D.: Commuting traces of upper triangular matrix rings. Aequ. Math. 91, 563–578 (2017)

    Article  MathSciNet  Google Scholar 

  17. Mathieu, M., Villena, A.R.: The structure of Lie derivations on \(C^*\)-algebras. J. Funct. Anal. 202, 504–525 (2003)

    Article  MathSciNet  Google Scholar 

  18. Martindale, M.S.: Lie derivations of primitive rings. Mich. Math. 11, 183–187 (1964)

    Article  MathSciNet  Google Scholar 

  19. Stenström B.: The maximal ring of quotients of a triangular matrix ring. Math. Scand. 34, 162–166 (1974)

    Article  MathSciNet  Google Scholar 

  20. Utumi, Y.: On quotient rings. Osaka J. Math. 8, 1–18 (1956)

    MathSciNet  MATH  Google Scholar 

  21. Villena, A.R.: Lie derivations on Banach algebras. J. Algebra 226, 390–409 (2000)

    Article  MathSciNet  Google Scholar 

  22. Wang, Y.: Functional identities of degree \(2\) in arbitrary triangular algebras. Linear Algebra Appl. 479, 171–184 (2015)

    Article  MathSciNet  Google Scholar 

  23. Wang, Y.: Biderivations of triangular rings. Linear Multilinear Algebra 64, 1952–1959 (2016)

    Article  MathSciNet  Google Scholar 

  24. Wang, Y.: Commuting (centralizing) traces and Lie (triple) isomorphisms on triangular algebras revisited. Linear Algebra Appl. 488, 45–70 (2016)

    Article  MathSciNet  Google Scholar 

  25. Zhang, J.H., Yu, W.Y.: Jordan derivations of triangular algebras. Linear Algebra Appl. 419, 251–255 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is grateful to the referee for useful comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yu Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Y. Lie (Jordan) derivations of arbitrary triangular algebras. Aequat. Math. 93, 1221–1229 (2019). https://doi.org/10.1007/s00010-018-0634-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00010-018-0634-8

Keywords

Mathematics Subject Classification

Navigation