Elementary Divisor Domains as Endomorphism Rings

  • László FuchsEmail author
Part of the Studies in Computational Intelligence book series (SCI, volume 878)


We consider modules over integral domains, and investigate direct sums of finite rank torsion-free modules whose endomorphism rings are completely integrally closed elementary divisor domains. The main result (Theorem 4.4) is a Krull-Schmidt-Azumaya type theorem: if a module A is a direct sum of finite rank torsion-free modules each of which is quasi-isomorphic to a member of a semi-rigid system with the mentioned type of endomorphism ring, then every summand of A admits the same kind of direct decomposition. The uniqueness of such decompositions is established up to quasi-isomorphism (Theorem 4.5).


Torsion-free Quasi-isomorphic modules Elementary divisor domains Endomorphism rings Semi-rigid systems Direct decompositions 

Mathematics Subject Classification (2010)

Primary 13C05 13G05 


  1. 1.
    Arnold, D., Hunter, R., Richman, F.: Global Azumaya theorems in additive categories. J. Pure Appl. Algebra 16, 223–242 (1980)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Azumaya, G.: Corrections and supplementaries to my paper concerning Krull-Remak-Schmidt’s theorem. Nagoya Math. J. 3, 117–124 (1950)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Botha, J.D., Gräbe, P.J.: On torsion-free abelian groups whose endomorphism rings are principal ideal domains. Comm. Algebra 11, 1343–1354 (1983)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brewer, J.W., Naudé, C., Naudé, G.: On Bézout domains, elementary divisor rings, and pole assignability. Comm. Algebra 12, 2987–3003 (1984)Google Scholar
  5. 5.
    Charles, B.: Sous-groupes fonctoriels et topologies. In: Studies of Abelian Groups. Dunod, Paris, pp. 72–92 (1968)Google Scholar
  6. 6.
    Corner, A.L.S.: Every countable reduced ring is an endomorphism ring. Proc. London Math. Soc. 13, 687–710 (1963)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Corner, A.L.S., Göbel, R.: Prescribing endomorphism algebras – a unified treatment. Proc. London Math. Soc. 50(3), 447–479 (1985)Google Scholar
  8. 8.
    Dugas, M., Göbel, R.: Every cotorsion-free algebra is an endomorphism algebra. Math. Z. 181, 451–470 (1982)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Facchini, A.: Krull-Schmidt and semilocal endomorphism rings. Lect. Notes Pure Appl. Math. Marcel Dekker, New York, 221, 181–201 (2001)Google Scholar
  10. 10.
    Fuchs, L.: Abelian Groups. Springer, Cham (2015)CrossRefGoogle Scholar
  11. 11.
    Fuchs, L., Salce, L.: Modules over non-Noetherian domains. Math. Surv. Monogr. 84 (2001). (Amer. Math. Society, Providence)Google Scholar
  12. 12.
    Goeters, P.: When summands of completely decomposable modules are completely decomposable. Comm. Algebra 35, 1956–1970 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kaplansky, I.: Projective modules. Ann. Math. 68, 372–377 (1958)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Warfield, R.: The Krull-Schmidt theorem for infinite sums of modules. Proc. Amer. Math. Soc. 22, 460–465 (1969)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsTulane UniversityNew OrleansUSA

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