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Hereditary algebras that are not pure-hereditary

  • Frank Okoh
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 832)

Abstract

Let R be a hereditary finite-dimensional algebra of tame type. In this note it is shown that any pure-injective R-module contains an indecomposable direct summand. This result is used to prove that if R is not countable and the category of R-modules contains a full sub-category equivalent to the category of representations of Ã1 then R is not pure-hereditary.

Keywords

Exact Sequence Direct Summand Hereditary Algebra Indecomposable Direct Summand Pure Submodule 
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.
    Fuchs, L.: Infinite abelian groups, Vol. I, Academic Press, New York and London, (1970).Google Scholar
  2. 2.
    Griffith, P.: On the decomposition of modules and generalized left uniserial rings, Math. Ann. 184 (1970) 300–308.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Jensen, C.U. and Simson D.: Purity and generalized chain conditions, Journal of Pure and Applied Algebra, 14 (1979) 297–305.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Okoh, F.: Direct sums and direct products of canonical pencils of matrices, Lin. Alg. and Appl. 25, (1979) 1–26.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    _____: Indecomposable pure-injective modules over finite-dimensional algebras of tame type, to appear.Google Scholar
  6. 6.
    Ringel, C.M.: Infinite-dimensional representations of finite-dimensional hereditary algebras, Symposia Math. Ins. Nat. Alta. Mat. 23 (1979).Google Scholar
  7. 7.
    Walker, C.P.: Relative homological algebra and abelian groups, Illinois J. Math. 10, (1966) 186–209.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Frank Okoh
    • 1
  1. 1.Department of MathematicsYork UniversityDownsviewCanada

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