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Journal of Mathematical Sciences

, Volume 202, Issue 3, pp 422–433 | Cite as

Serial Group Rings of Finite Groups. p-nilpotency

  • A. V. KukharevEmail author
  • G. E. Puninski
Article

It is proved that if F is an arbitrary field of characteristic p and G is a finite p-nilpotent group with a cyclic p-Sylow subgroup, then the group ring FG is serial. As a corollary, it is shown that for an arbitrary field F of characteristic 2 and any finite group G, the ring FG is serial if and only if the 2-Sylow subgroup of G is cyclic.

Keywords

Group Ring Chain Ring Artinian Ring Principal Ideal Ring Goldie Dimension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    J. L. Alperin, Local Representation Theory, Cambridge University Press (1986).Google Scholar
  2. 2.
    F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2d ed., Springer (1992).Google Scholar
  3. 3.
    Y. Baba and K. Oshiro, Classical Artinian Rings, World Scient. Publ. (2009).Google Scholar
  4. 4.
    C. W. Curtis and I. Reiner, Methods of Representation Theory with Applications to Finite Groups and Orders, Vol. 1, Wiley–Interscience (1981).Google Scholar
  5. 5.
    J. Gonzáles-Sánchez, “A p-nilpotency criterion,” Arch. Math., 94, 201–205 (2010).CrossRefGoogle Scholar
  6. 6.
    C. Faith, Algebra: Rings, Modules, and Categories [Russian translation], Vol. 2, Mir, Moscow (1979).Google Scholar
  7. 7.
    M. Hazewinkel, N. Gubareni, and V. V. Kirichenko, Algebras, Rings, and Modules, Vol. 1, Kluwer (2004).Google Scholar
  8. 8.
    D. G. Higman, “Indecomposable representations at characteristic p,” Duke J. Math., 21, 377–381 (1954).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    G. Hiss, C. Jansen, K. Lux, and P. Parker, Computational Modular Character Theory, http://www.math.rwth-aachen.de/homes/MOC/.
  10. 10.
    G. J. Janusz, “Indecomposable modules for finite groups,” Ann. Math., 89, 209–241 (1969).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    T. Y. Lam, Lectures on Modules and Rings, Springer (1999).Google Scholar
  12. 12.
    T. Y. Lam, A First Course in Noncommutative Rings, Springer (2001).Google Scholar
  13. 13.
    L. S. Levy and J. C. Robson, Hereditary Noetherian Prime Rings and Idealizers, AMS Math. Surveys and Monographs, Vol. 174 (2011).Google Scholar
  14. 14.
    K. Morita, “On group rings over a modular field which possess radicals expressible as principal ideals,” Sci. Repts. Tokyo Daigaku, 4, 177–194 (1951).zbMATHGoogle Scholar
  15. 15.
    I. Murase, “Generalized uniserial group rings. I,” Sci. Papers College Gener. Educ. Univ. Tokyo, 15, 15–28 (1965).MathSciNetzbMATHGoogle Scholar
  16. 16.
    I. Murase, “Generalized uniserial group rings. II,” Sci. Papers College Gener. Educ. Univ. Tokyo, 15, 111–128 (1965).MathSciNetzbMATHGoogle Scholar
  17. 17.
    D. S. Passman, The Algebraic Structure of Group Rings, Krieger Publishing Company (1985).Google Scholar
  18. 18.
    R. Pierce, Associative Algebras [Russian translation], Mir, Moscow (1986).Google Scholar
  19. 19.
    G. Puninski, Serial Rings, Kluwer (2001).Google Scholar
  20. 20.
    L. H. Rowen, Ring Theory, Academic Press (1991).Google Scholar
  21. 21.
    B. Srinivasan, “On the indecomposable representations of a certain class of groups,” Proc. Lond. Math. Soc., 10, 497–513 (1960).CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsBelarusian State UniversityMinskBelarus

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