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Bulletin of the Iranian Mathematical Society

, Volume 45, Issue 5, pp 1323–1338 | Cite as

The \(\pi \)-Semisimplicity of Locally Inverse Semigroup Algebras

  • Yingdan JiEmail author
Original Paper
  • 51 Downloads

Abstract

In this paper, we first characterize when a semigroup has completely 0-simple semigroup as its principal factors. Let R be a commutative ring with an identity, and let S be a locally inverse semigroup with the set of idempotents locally pseudofinite. Assume that the principal factors of S are all completely 0-simple. Then, we prove that the contracted semigroup algebra \(R_0[S]\) is \(\pi \)-semisimple if and only if the contracted semigroup algebras of all the principal factors of S are \(\pi \)-semisimple. Examples are provided to illustrate that the locally pseudofinite condition on the idempotent set of S cannot be removed. Notice that we extend the corresponding results on finite locally inverse semigroups.

Keywords

Locally inverse semigroup algebras Locally pseudofinite \(\pi \)-Semisimple Completely 0-simple semigroups Principal factors 

Mathematics Subject Classification

16G30 17C17 17C20 17C27 20M25 

Notes

References

  1. 1.
    Amitsur, S.A.: A general theory of radicals. II. Radicals in rings and bicategories. Am. J. Math. 76(1), 100–125 (1954)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroup. Mathematical Surveys, vol. 7. American Mathematical Society, Providence (1961)zbMATHGoogle Scholar
  3. 3.
    Domanov, A.I.: On semisimplicity and identities of inverse semigroup algebras. Mat. Issled. 38(207), 123–137 (1976) (Russian) Google Scholar
  4. 4.
    Green, J.A.: On the structure of semigroups. Ann. Math. 54, 163–172 (1951)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Guo, X.J.: A note on locally inverse semigroup algebras. Int. J. Math. Math. Sci. 2008, 576061 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Howie, J.M.: Fundamentals of Semigroup Theory. Oxford University Press, New York (1995)zbMATHGoogle Scholar
  7. 7.
    Ji, Y.D., Luo, Y.F.: Locally adequate semigroup algebras. Open Math. 14, 29–48 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ji, Y.D., Luo, Y.F.: Semiprimitivity of orthodox semigroup algebras. Commun. Algebra 44(12), 5149–5162 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kelarev, A.V.: Ring Constructions and Applications. World Scientific, New Jersey (2002)zbMATHGoogle Scholar
  10. 10.
    Munn, W.D.: A class of contracted inverse semigroup rings. Proc. R. Soc. Edinb. 107A(1–2), 175–196 (1987)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Okninski, J.: Semigroup Algebras. M. Dekker, New York (1991)zbMATHGoogle Scholar
  12. 12.
    Ponizovskĭ, I.S.: On the semiprimitivity of inverse semigroup algebras and on theorems by Domanov and Munn. Semigroup Forum 40, 181–185 (1990)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Rukolaĭne, A.V.: Semigroup algebras of finite inverse semigroups over arbitrary fields. J. Math. Sci. 24(4), 460–464 (1984)CrossRefGoogle Scholar
  14. 14.
    Shojaee, B., Esslamzadeh, G.H., Pourabbas, A.: First order cohomology of \(\ell ^1\)-Munn algebras and certain semigroup algebras. Bull. Iran. Math. Soc. 35, 211–219 (2009)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Steinberg, B.: Möbius functions and semigroup representation theory. J. Comb. Theory A 113, 866–881 (2006)CrossRefGoogle Scholar
  16. 16.
    Teply, M.L., Turman, E.G., Quesada, A.: On semisimple semigroup rings. Proc. Am. Math. Soc. 79, 157–163 (1980)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Weissglass, J.: Radicals of semigroup rings. Glasg. Math. J. 10, 85–93 (1969)MathSciNetCrossRefGoogle Scholar

Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.School of Applied MathematicsGuangdong University of TechnologyGuangzhouPeople’s Republic of China

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