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Positivity

, Volume 18, Issue 3, pp 603–617 | Cite as

Higher derivations and commutativity in lattice-ordered rings

  • S. Andima
  • H. PajooheshEmail author
Article

Abstract

In 1978 I. N. Herstein proved that a prime ring \(R\) of characteristic not two with nonzero derivation \(d\) satisfying \(d(x)d(y)=d(y)d(x)\) for all \(x,y\in R\) is commutative, and in 1995 Bell and Daif showed that \(d(xy)=d(yx)\) implies commutativity. We extend the Bell–Daif theorem to lattice-ordered prime rings with a positive derivation satisfying the property on a one-sided \(L\)-ideal and interpret these conditions for higher derivations in prime \(d\)-rings and in semiprime \(f\)-rings. Our key tool is that every positive derivation nilpotent on a one-sided \(L\)-ideal of a semiprime \(\ell \)-ring is zero on that ideal.

Keywords

Derivation Higher derivation Lattice-ordered ring \(f\)-ring \(d\)-ring Prime ring Semiprime ring 

Mathematics Subject Classification (2000)

16N60 13N15 16U80 06F25 

References

  1. 1.
    Andima, S., Pajoohesh, H.: Commutativity of rings with derivations. Acta Math. Hungar. 128, 1–14 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bell, H.E., Daif, M.N.: On derivations and commutativity in prime rings. Acta Math. Hungar. 66, 337–343 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bigard, A., Keimel, K., Wolfenstein, S.: Groupes et Anneaux Reticules. Lecture Notes in Mathematics Series, vol. 608. Springer, Berlin (1977)Google Scholar
  4. 4.
    Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society Colloquium Publications, vol. 25. American Mathematical Society, Providence (1967)Google Scholar
  5. 5.
    Chung, L.O., Luh, J.: Nilpotency of derivations on an ideal. Proc. Am. Math. Soc. 90, 211–214 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Colville, P., Davis, G., Keimel, K.: Positive derivations on \(f\)-rings. J Aust. Math. Soc. Ser. A 23, 371–375 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Darnel, M.: Theory of Lattice-Ordered Groups. Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, New York (1995)Google Scholar
  8. 8.
    DeMarr, R.: On partially ordering operator algebras. Canad. J. Math. 19, 636–643 (1967)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Ebrahimi, M., Pajoohesh, H.: Composition of derivations on (semi)prime \(\ell \)-rings. H. Kyungpook Math. J. 44, 293–297 (2004)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Henriksen, M., Smith, F.A.: Some properties of positive derivations on f-rings. Contemp. Math. 8, 175–184 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Herstein, I.N.: A note on derivations. Canad. Math. Bull. 21, 369–370 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Hungerfor, T.W.: Algebra. Springer, Berlin (1974)Google Scholar
  13. 13.
    Jacobson, N.: Structure theory for algebraic algebras of bounded degree. Ann. Math. (2) 46, 695–707 (1945)Google Scholar
  14. 14.
    Lam, T.Y.: A first course in noncommutative rings. In: Graduate Texts in Mathematics, vol. 131. Springer, New York (1991)Google Scholar
  15. 15.
    Mayne, J.H.: Ideals and centralizing mappings in prime rings. Proc. Am. Math. Soc. 86, 211–212 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Pajoohesh, H.: Lattice-ordered rings and derivations. Thesis submitted to Shahid Beheshti University (2003)Google Scholar
  17. 17.
    Posner, E.C.: Derivations in prime rings. Proc. Am. Math. Soc. 8, 1093–1100 (1957)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Steinberg, S.: Lattice-Ordered Rings and Modules. Springer, New York (2010)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsLong Island University-C.W. Post CampusBrookvilleUSA
  2. 2.Department of MathematicsMedgar Evers College of CUNYBrooklynUSA

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