Advertisement

Mediterranean Journal of Mathematics

, Volume 8, Issue 3, pp 271–291 | Cite as

Generalized Jordan Derivations on Semiprime Rings and Its Applications in Range Inclusion Problems

  • Feng Wei
  • Zhankui Xiao
Article

Abstract

It is shown that any generalized Jordan (triple-)derivation on a 2–torsion free semiprime ring is a generalized derivation and that any generalized Jordan higher derivation on a 2–torsion free semiprime ring is a generalized higher derivation. Then we give several conditions which enable some generalized Jordan derivations on prime rings to degenerate left or right multipliers. Lastly, we apply these degenerating conditions to discuss the range inclusion problems of generalized derivations on noncommutative Banach algebras.

Mathematics Subject Classification (2010)

Primary 16W25 16N60 Secondary 47B47 

Keywords

Generalized Jordan derivation generalized Jordan triple derivation (semi–)prime ring Banach algebra 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Albas N., Albas E.: Generalized derivations of prime rings Algebra Colloq. 11, 399–410 (2004)MathSciNetMATHGoogle Scholar
  2. 2.
    Argac N., Albas E.: On generalized (σ, τ)-derivations. Siberian Math. J. 43, 977–984 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beidar K.I.: Rings of quotients of semiprime rings. VestnikMoskov. Univ. Ser I Mat. Meh. (Engl. Transl. Moscow Univ. Math. Bull.) 33, 36–42 (1978)Google Scholar
  4. 4.
    Brešar M.: Jordan mappings of semiprime rings. J. Algebra 127, 218–228 (1989)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Brešar M., Mathieu M.: Derivations mapping into the radical. J. Funct. Anal. 133, 21–29 (1995)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Cusack J.M.: Jordan derivations on rings. Proc. Amer. Math. Soc. 53, 321–324 (1975)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    De Filippis V.: An Engel condition with generalized derivations on multilinear polynomials. Israel J. Math. 162, 93–108 (2007)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    De Filippis V.: Posner’s second theorem and an annihilator condition with generalized derivations. Turkish J. Math. 32, 197–211 (2008)MathSciNetMATHGoogle Scholar
  9. 9.
    De Filippis V.: Generalized derivations in prime rings and noncommutative Banach algebras. Bull. Korean Math. Soc. 45, 621–629 (2008)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    De Filippis V.: Annihilators of power values of generalized derivations on multilinear polynomial. Bull. Aust. Math. Soc. 80, 217–232 (2009)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    De Filippis V., Inceboz H.G.: Generalized derivations with power central values on multilinear polynomials on right ideals. Rend. Sem. Mat. Univ. Padova 120, 59–71 (2008)MATHGoogle Scholar
  12. 12.
    De Filippis V., Tammam El-Sayiad M.S.: A note on Posner’s theorem with generalized derivations on Lie ideals. Rend. Sem. Mat. Univ. Padova 122, 55–64 (2009)MATHGoogle Scholar
  13. 13.
    Ferrero M., Haetinger C.: Higher derivations and a theorem by Herstein. Quaest. Math. 25, 249–257 (2002)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hvala B.: Generalized derivations in rings. Comm. Algebra 26, 1147–1166 (1998)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Jing W., Lu S.-J.: Generalized Jordan derivations on prime rings and standard operator algebras. Taiwanese J. Math. 7, 605–613 (2003)MathSciNetMATHGoogle Scholar
  16. 16.
    Johnson B.E., Sinclair A.M.: Continuity of derivations and a problem of Kaplansky. Amer. J. Math. 90, 1067–1073 (1968)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Jung Y.-S.: Generalized Jordan triple higher derivations on prime rings. Indian J. Pure Appl. Math. 36, 513–524 (2005)MathSciNetMATHGoogle Scholar
  18. 18.
    Lee T.-K.: Semiprime rings with differential identities. Bull. Inst. Math. Acad. Sinica 20, 27–38 (1992)MathSciNetMATHGoogle Scholar
  19. 19.
    Mathieu M., Runde V.: Derivations mapping into the radical, II. Bull. London Math. Soc. 24, 485–487 (1992)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Nakajima A.: On generalized higher derivations. Turk. J. Math. 24, 295–311 (2000)MathSciNetMATHGoogle Scholar
  21. 21.
    Sinclair A.M.: Jordan homomorphisms and derivations on semisimple Banach algebras. Proc. Amer. Math. Soc. 24, 209–214 (1970)MathSciNetMATHGoogle Scholar
  22. 22.
    A. M. Sinclair, Automatic Continuity of Linear Operators, London Mathematical Society Lecture Note Series, V. 21, Cambridge University Press, Cambridge, 1976.Google Scholar
  23. 23.
    Singer I.M., Wermer J.: Derivations on commutative normed algebras. Math. Ann. 129, 260–264 (1955)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Thomas M.P.: The image of a derivation is contained in the radical. Ann. Math. 128, 435–460 (1993)CrossRefGoogle Scholar
  25. 25.
    Thomas M.P.: Primitive ideals and derivations on noncommutative Banach algebras. Pacific. J. Math. 159, 139–152 (1993)MathSciNetMATHGoogle Scholar
  26. 26.
    Wei F.: Generalized derivations with nilpotent values on semiprime rings. Acta Math. Sinica 20, 453–462 (2004)MATHCrossRefGoogle Scholar
  27. 27.
    Wei F.: Generalized differential identities of (semi-)prime rings. Acta Math. Sinica 21, 823–832 (2005)MATHCrossRefGoogle Scholar
  28. 28.
    Wei F., Xiao Z.-K.: Generalized Jordan derivations on semiprime rings. Demonstratio Math. 40, 789–798 (2007)MathSciNetMATHGoogle Scholar
  29. 29.
    Wei F., Xiao Z.-K.: Generalized Jordan triple higher derivations on semiprime rings. Bull. Korean Math. Soc. 46, 553–565 (2009)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Wei F., Xiao Z.-K.: Generalized derivetions on (semi-)prime rings and noncommutative Banach algebras. Rend. Sem. Mat. Univ. Padova 122, 171–190 (2009)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of Applied MathematicsBeijing Institute of TechnologyBeijingP. R. China

Personalised recommendations