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


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 


Prime and semiprime ring generalized derivation 


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  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

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