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Lie (Jordan) derivations of arbitrary triangular algebras

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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.

Keywords

Triangular algebra Maximal left ring of quotients Lie derivation Jordan derivation Derivation 

Mathematics Subject Classification

16W25 16R60 

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Notes

Acknowledgements

The author is grateful to the referee for useful comments.

References

  1. 1.
    Beidar, K.I., Chebotar, M.A.: On Lie derivations of Lie ideals of prime rings. Isr. J. Math. 123, 131–148 (2001)CrossRefGoogle Scholar
  2. 2.
    Beidar, K.I., Martindale 3rd, W.S., Mikhale, A.: Rings with Generalized Identities. Marcel Dekker, NewYork (1996)Google Scholar
  3. 3.
    Benkovič, D.: Lie triple derivations of unital algebras with idempotents. Linear Multilinear Algebra 63, 141–165 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Benkovič, D.: Lie derivations on triangular matrices. Linear Multilinear Algebra 55, 619–629 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Benkovič, D.: Generalized Lie derivations on triangular rings. Linear Algebra Appl. 434, 1532–1544 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Benkovič, D.: A note on \(f\)-derivations of triangular algebras. Aequ. Math. 89, 1207–1211 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Benkovič, D., Eremita, D.: Commuting traces and commutativity preserving maps on triangular algebras. J. Algebra 280, 797–824 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Benkovič, D., Eremita, D.: Multiplicative Lie \(n\)-derivations of triangular rings. Linear Algebra Appl. 436, 4223–4240 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Brešar, M.: Commuting traces of biadditive mappings, commutativity preserving mappings and Lie mappings. Trans. Am. Math. Soc. 335, 525–546 (1993)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Brešar, M., Chebotar, M.A.: Martindale 3rd W. S.: Functional Identities. Birkhäuser Verlag, Basel (2007)Google Scholar
  11. 11.
    Brešar, M., Šemel, P.: Commuting traces of biadditive maps revisited. Commun. Algebra 31, 381–388 (2003)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cheung, W.S.: Commuting maps of triangular algebras. J. Lond. Math. Soc. 63, 117–127 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cheung, W.S.: Lie derivations of triangular algebras. Linear Multilinear Algebra 51, 299–310 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Eremita, D.: Functional identities of degree \(2\) in triangular rings. Linear Algebra Appl. 438, 584–597 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Eremita, D.: Functional identities of degree \(2\) in triangular rings revisited. Linear Multilinear Algebra 63, 534–553 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Eremita, D.: Commuting traces of upper triangular matrix rings. Aequ. Math. 91, 563–578 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mathieu, M., Villena, A.R.: The structure of Lie derivations on \(C^*\)-algebras. J. Funct. Anal. 202, 504–525 (2003)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Martindale, M.S.: Lie derivations of primitive rings. Mich. Math. 11, 183–187 (1964)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Stenström B.: The maximal ring of quotients of a triangular matrix ring. Math. Scand. 34, 162–166 (1974)Google Scholar
  20. 20.
    Utumi, Y.: On quotient rings. Osaka J. Math. 8, 1–18 (1956)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Villena, A.R.: Lie derivations on Banach algebras. J. Algebra 226, 390–409 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Wang, Y.: Functional identities of degree \(2\) in arbitrary triangular algebras. Linear Algebra Appl. 479, 171–184 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wang, Y.: Biderivations of triangular rings. Linear Multilinear Algebra 64, 1952–1959 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Wang, Y.: Commuting (centralizing) traces and Lie (triple) isomorphisms on triangular algebras revisited. Linear Algebra Appl. 488, 45–70 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Zhang, J.H., Yu, W.Y.: Jordan derivations of triangular algebras. Linear Algebra Appl. 419, 251–255 (2006)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiChina

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