Algebras and Representation Theory

, Volume 21, Issue 2, pp 309–329 | Cite as

The Krull-Schmidt Theorem Holds for Finite Direct Products of Biuniform Groups

Article
  • 37 Downloads

Abstract

We prove that the Krull-Schmidt Theorem holds for finite direct products of biuniform groups, that is, groups G whose lattice of normal subgroups \(\mathcal {N}(G)\) has Goldie dimension and dual Goldie dimension 1. More generally, it holds for the class of completely indecomposable groups.

Keywords

Lattice of normal subgroups Krull-Schmidt Theorem Normal homomorphism 

Mathematics Subject Classification (2010)

20E15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

We are grateful to the referee for a careful reading of the previous versions of the paper.

References

  1. 1.
    Amini, B., Amini, A., Facchini, A.: Equivalence of diagonal matrices over local rings. J. Algebra 320, 1288–1310 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Anderson, F.W., Fuller, K.R.: Rings and categories of modules, 2nd ed, GTM, vol. 13. Springer-Verlag, New York (1992)CrossRefGoogle Scholar
  3. 3.
    Ashmanov, I.S., Ol’shanskii, A.Yu.: Abelian and central extensions of aspherical groups. Izv. Vyssh. Uchebn. Zaved. Mat. 11(85), 48–60 (1985)MathSciNetGoogle Scholar
  4. 4.
    Azumaya, G.: Corrections and supplementaries to my paper concerning Krull-Remak-Schmidt’s theorem. Nagoya Math. J. 1, 117–124 (1950)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Crawley, P., Jónsson, B.: Refinements for infinite direct decompositions of algebraic systems. Pac. J. Math. 14, 797–855 (1964)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ecevit, Ş., Facchini, A., Koşan, M.T.: Direct sums of infinitely many kernels. J. Austral. Math. Soc. 89, 199–214 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Facchini, A.: Krull-Schmidt fails for serial modules. Trans. Amer. Math. Soc. 348, 4561–4575 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Facchini, A.: Module Theory. Endomorphism rings and direct sum decompositions in some classes of modules. Progress in Mathematics 167. Birkhäuser Verlag, Basel (1998). Reprinted in Modern Birkhäuser Classics, Birkhäuser Verlag, BaselMATHGoogle Scholar
  9. 9.
    Facchini, A.: Injective modules, spectral categories, and applications. In: Jain, S.K., Parvathi, S. (eds.) Noncommutative Rings, Group Rings, Diagram Algebras and Their Applications, pp 1–17. Contemporary Mathematics 456. American Mathematics Society (2008)Google Scholar
  10. 10.
    Facchini, A.: Direct-sum decompositions of modules with semilocal endomorphism rings. Bull. Math. Sci. 2, 225–279 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Facchini, A., Ecevit, Ş., Koşan, M.T.: Kernels of morphisms between indecomposable injective modules. Glasg. Math. J. 52A, 69–82 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Facchini, A., Girardi, N.: Couniformly presented modules and dualities. In: Van Huynh, D., López-Permouth, S.R. (eds.) Advances in Ring Theory, pp 149–164. Trends in Mathematics, Birkhäuser Verlag, Basel (2010)Google Scholar
  13. 13.
    Facchini, A., Herbera, D.: Local morphisms and modules with a semilocal endomorphism ring. Algebr. Represent. Theory 9, 403–422 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Facchini, A., Prh́oda, P.: Endomorphism rings with finitely many maximal right ideals. Comm. Algebra 39, 3317–3338 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fuchs, L.: Infinite Abelian Groups, Vol. II, Pure and Applied Mathematics, vol. 36-II. Academic Press, New York-London (1973)Google Scholar
  16. 16.
    Gabriel, P., Oberst, U.: Spektralkategorien und reguläre Ringe im Von-Neumannschen Sinn. Math. Zeitschr. 82, 389–395 (1966)CrossRefMATHGoogle Scholar
  17. 17.
    Girardi, N.: Regular Biproduct Decompositions of Objects. PhD thesis, Università di Padova (2011)Google Scholar
  18. 18.
    Grzeszczuk, P., Puczyowski, E.R.: On Goldie and dual Goldie dimensions. J. Pure Appl. Algebra 31, 47–54 (1984)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hirshon, R.: On cancellation in groups. Amer. Math. Mon. 76, 1037–1039 (1969)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Hirshon, R.: Cancellation of groups with maximal condition. Proc. Amer. Math. Soc. 24, 401–403 (1970)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Hirshon, R.: Some cancellation theorems with applications to nilpotent groups. J. Austral. Math. Soc. Ser. A 23(2), 147–165 (1977)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Meier, D.: Non-hopfian groups. J. London Math. Soc. 26, 265–270 (1982)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Ol’shanskii, A.: Geometry of defining relations in groups, Translated from the 1989 Russian original by Yu. A. Bakhturin Mathematics and its Applications (Soviet Series), vol. 70. Kluwer Academic Publishers Group, Dordrecht (1991)Google Scholar
  24. 24.
    Robinson, D.J.S.: A course in the theory of groups. 2nd edn, GTM, vol. 80. Springer, New York (1996)CrossRefGoogle Scholar
  25. 25.
    Stenström, B.: Rings of quotients. Springer-Verlag, Berlin-Heidelberg, New York (1975)CrossRefMATHGoogle Scholar
  26. 26.
    Tyrer Jones, J.M.: Direct products and the Hopf property. J. Austral. Math Soc. 27, 174–198 (1974)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Walker, E.A.: Cancellation in direct sums of groups. Proc. Amer. Math. Soc. 7, 898–902 (1956)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Warfield, R.B. Jr: Serial rings and finitely presented modules. J. Algebra 37, 187–222 (1975)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPadovaItaly

Personalised recommendations