Advertisement

Aequationes mathematicae

, Volume 85, Issue 3, pp 529–537 | Cite as

On one sided ideals of a semiprime ring with generalized derivations

  • Asma Ali
  • Vincenzo De Filippis
  • Faiza Shujat
Article
  • 218 Downloads

Abstract

Let R be a ring with center Z(R). An additive mapping \({F : R \longrightarrow R}\) is said to be a generalized derivation on R if there exists a derivation \({d : R \longrightarrow R}\) such that F(xy) = F(x)y + xd(y), for all \({x, y \in R}\) (the map d is called the derivation associated with F). Let R be a semiprime ring and U be a nonzero left ideal of R. In the present note we prove that if R admits a generalized derivation F, d is the derivation associated with F such that d(U) ≠ (0) then R contains some nonzero central ideal, if one of the following conditions holds: (1) R is 2-torsion free and \({F(xy) \in Z(R)}\), for all \({x, y \in U}\), unless F(U)U = UF(U) = Ud(U) = (0); (2) \({F(xy) \mp yx \in Z(R)}\), for all \({x,y \in U}\); (3) \({F(xy) \mp [x,y] \in Z(R)}\), for all \({x,y \in U}\); (4) F ≠ 0 and F([x,y]) = 0, for all \({x, y \in U}\), unless Ud(U) = (0); (5) F ≠ 0 and \({F([x, y]) \in Z(R)}\), for all \({x, y \in U}\), unless either d(Z(R))U = (0) or Ud(U) = (0)n.

Mathematics Subject Classification (2000)

16W25 16W20 16N60 

Keywords

Prime and semiprime ring generalized derivation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brešar M.: Centralizing mappings and derivations in prime rings. J. Algebra 156, 385–394 (1993)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Daif M.N., Bell H.E.: Remarks on derivations on semiprime rings. Int. J. Math. Math. Sci. 15(1), 205–206 (1992)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Dhara, B.: Remarks on generalized derivations in prime and semiprime rings. Int. J. Math. Math. Sci. ID 646587 (2010)Google Scholar
  4. 4.
    Fošner A., Fošner M., Vukman J.: Identities with derivations in rings. Glas. Mat. 46, 339–349 (2011)MATHCrossRefGoogle Scholar
  5. 5.
    Fošner A., Vukman J.: Some results concerning additive mappings and derivations on semiprime rings. Publ. Math. Debrecen 78, 575–581 (2011)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Fošner A., Vukman J.: On certain functional equations related to Jordan triple \({(\vartheta, \varphi)}\) derivations on semiprime rings. Monatsh. Math. 162, 157–165 (2011)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Herstein I.N.: Rings with Involution. University of Chicago Press, Chicago (1976)MATHGoogle Scholar
  8. 8.
    Hvala B.: Generalized derivations in rings. Commun. Algebra 26, 1147–1166 (1998)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Lam T.Y.: A First Course in Noncommutative Rings. Graduate Texts in Mathematics. Springer, Berlin (2001)CrossRefGoogle Scholar
  10. 10.
    Lanski C.: An Engel condition with derivation for left ideals. Proc. Am. Math. Soc. 125(2), 339–345 (1997)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Lanski C.: Left ideals and derivations in semiprime rings. J. Algebra 277, 658–667 (2004)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Lee T.K.: Generalized derivations of left faithful rings. Commun. Algebra 27(8), 4057–4073 (1999)MATHCrossRefGoogle Scholar
  13. 13.
    Quadri M.A., Khan M.S., Rehman N.: Generalized derivations and commutativity of prime rings. Indian J. Pure Appl. Math. 34(9), 1393–1396 (2003)MathSciNetMATHGoogle Scholar
  14. 14.
    Vukman J.: A note on generalized derivations of semiprime rings. Taiwan J. Math. 11, 367–370 (2007)MathSciNetMATHGoogle Scholar
  15. 15.
    Zalar B.: On centralizers of semiprime rings. Comment. Math. Univ. Carol. 32(4), 609–614 (1991)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of MathematicsAligarh Muslim UniversityAligarhIndia
  2. 2.DI.S.I.A., Faculty of EngineeringUniversity of MessinaMessinaItaly

Personalised recommendations