Israel Journal of Mathematics

, Volume 72, Issue 1–2, pp 5–18 | Cite as

Irreducible actions and faithful actions of hopf algebras

  • Jeffrey Bergen
  • Miriam Cohen
  • Davida Fischman


LetH be a Hopf algebra acting on an algebraA. We will examine the relationship betweenA, the ring of invariantsA H, and the smash productA # H. We begin by studying the situation whereA is an irreducibleA # H module and, as an application of our first main theorem, show that ifD is a division ring then [D : D H]≦dimH. We next show that prime rings with central rings of invariants satisfy a polynomial identity under the action of certain Hopf algebras. Finally, we show that the primeness ofA # H is strongly related to the faithfulness of the left and right actions ofA # H onA.


Hopf Algebra Left Ideal Prime Ring Division Ring Polynomial Identity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A 80]
    E. Abe,Hopf Algebras, Cambridge Univ. Press, Cambridge, 1980.MATHGoogle Scholar
  2. [B 88]
    J. Bergen,Constants of Lie algebra actions, J. Algebra114 (1988), 452–465.MATHCrossRefMathSciNetGoogle Scholar
  3. [B 89]
    J. Bergen,Automorphic-differential identities in rings, Proc. Am. Math. Soc.106 (2) (1989), 297–305.MATHCrossRefMathSciNetGoogle Scholar
  4. [BI 73]
    G. M. Bergman and I. M. Isaacs,Rings with fixed-point-free group actions, Proc. London Math. Soc27 (1973), 69–87.MATHCrossRefMathSciNetGoogle Scholar
  5. [BM 86]
    J. Bergen and S. Montgomery,Smash products and outer derivations, Isr. J. Math.53 (1986), 321–345.MATHMathSciNetGoogle Scholar
  6. [Br]
    A. Braun, unpublished.Google Scholar
  7. [C 87]
    M. Cohen,H-Simple H-module algebras, Canadian Math. Bull.30(3) (1987), 363–366.MATHGoogle Scholar
  8. [CF 86]
    M. Cohen and D. Fischman,Hopf algebra actions, J. Algebra100 (1986), 363–379.MATHCrossRefMathSciNetGoogle Scholar
  9. [CFM 90]
    M. Cohen, D. Fischman and S. Montgomery,Hopf Galois extensions, smash products, and Morita equivalence, J. Algebra133 (1990), 351–372.MATHCrossRefMathSciNetGoogle Scholar
  10. [HN 75]
    I. N. Herstein and L. Neumann,Centralizers in rings, Ann. Mat. Pura Appl.102(4) (1975), 37–44.MathSciNetGoogle Scholar
  11. [J 56]
    N. Jacobson,Structure of Rings, Am. Math. Soc. Colloquium Publ., 1956.Google Scholar
  12. [J 62]
    N. Jacobson,Lie Algebras, Wiley-Interscience, New York, 1962.MATHGoogle Scholar
  13. [K 74]
    V. K. Kharchenko,Galois extensions and rings of quotients, Algebra i Logika13(4) (1974), 460–484, 488 (English transl. (1975), 265–281).MATHMathSciNetGoogle Scholar
  14. [L 66]
    J. Lambek,Rings and Modules, Blaisdell, Waltham, Massachusetts, 1966.MATHGoogle Scholar
  15. [M 69]
    W. Martindale,Prime rings satisfying a generalized polynomial identity, J. Algebra12 (1969), 576–584.MATHCrossRefMathSciNetGoogle Scholar
  16. [McS 71]
    J. C. McConnell and M. E. Sweedler,Simplicity of smash products, Proc. London Math. Soc.23(3) (1971), 251–266.MATHCrossRefMathSciNetGoogle Scholar
  17. [O 81]
    J. Osterburg,Central fixed rings, J. London Math. Soc.23(2) (1981), 246–248.MATHCrossRefMathSciNetGoogle Scholar
  18. [P 83]
    A. Z. Popov,Derivations of prime rings, Algebra i Logika22 (1983), 79–92.MathSciNetGoogle Scholar
  19. [S 68]
    M. E. Sweedler,Cohomology of algebras over Hopf algebras, Trans. Am. Math. Soc.133 (1968), 203–239.CrossRefMathSciNetGoogle Scholar
  20. [S 69a]
    M. E. Sweedler,Hopf Algebras, Benjamin, New York, 1969.Google Scholar
  21. [S 69b]
    M. E. Sweedler,Integrals for Hopf algebras, Ann. of Math.89(2) (1969), 323–335.CrossRefMathSciNetGoogle Scholar

Copyright information

© The Weizmann Science Press of Israel 1990

Authors and Affiliations

  • Jeffrey Bergen
    • 1
  • Miriam Cohen
    • 2
  • Davida Fischman
    • 3
  1. 1.DePaul UniversityChicagoUSA
  2. 2.Ben Gurion University of the NegevBeer ShevaIsrael
  3. 3.Weizmann Institute of ScienceRehovotIsrael

Personalised recommendations