Skip to main content
SpringerLink
Log in

On multiplicative (generalized)-derivations in prime and semiprime rings

  • Published:
Aequationes mathematicae Aims and scope Submit manuscript

Abstract

Let R be a ring. A map \({F : R \rightarrow R}\) is called a multiplicative (generalized)-derivation if F(xy) = F(x)y + xg(y) is fulfilled for all \({x, y \in R}\) where \({g : R \rightarrow R}\) is any map (not necessarily derivation). The main objective of the present paper is to study the following situations: (i) \({F(xy) \pm xy \in Z}\), (ii) \({F(xy) \pm yx \in Z}\), (iii) \({F(x)F(y) \pm xy \in Z}\) and (iv) \({F(x)F(y) \pm yx \in Z}\) for all x, y in some appropriate subset of R. Moreover, some examples are also given.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ali, S., Dhara, B., Fošner, A.: Some commutativity theorems concerning additive maps and derivations on semiprime rings. Proceedings of 6th China–Japan–Korea international conference on Ring theory. In: Kwak, T.K., Lee, Y. (eds.), pp. 135–143, World Scientific, Singapore (2012)

  2. Anderson F.W.: Lectures on noncommutative rings. University of Oregon, Oregon (2002)

    Google Scholar 

  3. Andima S., Pajoohesh H.: Commutativity of rings with derivations. Acta Math. Hung. 128(1–2), 1–14 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ashraf M., Ali A., Ali S.: Some commutativity theorems for rings with generalized derivations. Southeast Asian Bull. Math. 31, 415–421 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Ashraf M., Rehman N.: On derivations and commutativity in prime rings. East-West J. Math. 3(1), 87–91 (2001)

    MathSciNet  MATH  Google Scholar 

  6. Bell H.E., Martindale W.S. III.: Centralizing maps of semiprime rings. Can. Math. Bull. 30(1), 92–101 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bell H.E., Daif M.N.: On derivations and commutativity in prime rings. Acta Math. Hung. 66(4), 337–343 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brešar, M.: On the distance of the composition of two derivations to the generalized derivations. Glasgow Math. J. (33), 89–93 (1991)

    Google Scholar 

  9. Brešar M.: Commuting maps: A survey. Taiwanese J. Math. 8(3), 361–397 (2004)

    MathSciNet  MATH  Google Scholar 

  10. Daif M.N.: When is a multiplicative derivation additive. Int. J. Math. Math. Sci. 14(3), 615–618 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Daif M.N., Bell H.E.: Remarks on derivations on semiprime rings. Int. J. Math. Math. Sci. 15(1), 205–206 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Daif M.N., Tammam El-Sayiad M.S.: Multiplicative generalized derivations which are additive. East-West J. Math. 9(1), 31–37 (1997)

    MathSciNet  Google Scholar 

  13. Dhara, B., Pattanayak, A.: Generalized derivations and left ideals in prime and semiprime rings. ISRN Algebra (2011) Article ID 750382, 5 pages

  14. Dhara, B.: Remarks on generalized derivations in prime and semiprime rings. Int. J. Math. Math. Sci. Volume 2010, Article ID 646587, 6 pages

  15. Eremita D., Ilisevic D.: On additivity of centralizers. Bull. Aust. Math. Soc. 74, 177–184 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Goldmann H., Šemrl P.: Multiplicative derivations on C(X). Monatsh. Math. 121(3), 189–197 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Herstein I.N.: Rings with involution. The University of Chicago Press, Chicago (1976)

    MATH  Google Scholar 

  18. Hvala B.: Generalized derivations in rings. Comm. Algebra 26(4), 1147–1166 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Martindale W.S. III: When are multiplicative maps additive. Proc. Amer. Math. Soc. 21, 695–698 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  20. Marubayashi M., Ashraf M., Rehman N., Ali S.: On generalized (α, β)-derivations in prime rings. Algebra Colloq. 17(Spec 1), 865–874 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Posner E.C.: Derivation in prime rings. Proc. Amer. Math. Soc. 8, 1093–1100 (1957)

    Article  MathSciNet  Google Scholar 

  22. 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)

    MathSciNet  MATH  Google Scholar 

  23. Rehman N.: On commutativity of rings with generalized derivations. Math. J. Okayama Univ. 44, 43–49 (2002)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Belda College, Belda, Paschim Medinipur, 721424, India

    Basudeb Dhara

  2. Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

    Shakir Ali

Authors
  1. Basudeb Dhara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shakir Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shakir Ali.

Additional information

This research is partially supported by the research grants from UGC, India (Grant No. F. PSW-099/10-11 and 39-37/2010(SR), respectively).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dhara, B., Ali, S. On multiplicative (generalized)-derivations in prime and semiprime rings. Aequat. Math. 86, 65–79 (2013). https://doi.org/10.1007/s00010-013-0205-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00010-013-0205-y

Mathematics Subject Classification (2000)

Keywords

Search

Navigation