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Siberian Mathematical Journal

, Volume 42, Issue 5, pp 987–990 | Cite as

On Purity and Quasiequality of Abelian Groups

  • M. A. Turmanov
Article
  • 13 Downloads

Abstract

We study the Cohn purity in an abelian group regarded as a left module over its endomorphism ring. We prove that if a finite rank torsion-free abelian group G is quasiequal to a direct sum in which all summands are purely simple modules over their endomorphism rings then the module E(G)G is purely semisimple. This theorem makes it possible to construct abelian groups of any finite rank which are purely semisimple over their endomorphism rings and it reduces the problem of endopure semisimplicity of abelian groups to the same problem in the class of strongly indecomposable abelian groups.

Keywords

Abelian Group Simple Module Endomorphism Ring Finite Rank Left Module 
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

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    Fuchs L., Infinite Abelian Groups. Vol. 1 and 2 [Russian translation], Nauka, Moscow (1974, 1977).Google Scholar
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    Mishina A. P. and Skornyakov L. A., Abelian Groups and Modules. Vol. 1 and 2 [in Russian], Nauka, Moscow (1969, 1977).Google Scholar
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    Cohn P., “On the free product of associative rings. I,” Math. Z., Bd 71, 380-398 (1959).Google Scholar
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    Arnold D., “Finite rank torsion free abelian groups and rings,” Lecture Notes in Math., 931 (1982).Google Scholar
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    Turmanov M. A., Endopure Submodules of Abelian Groups [in Russian], Avtoref. Dis. Kand. Fiz.-Mat. Nauk, Moscow (1991).Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • M. A. Turmanov
    • 1
  1. 1.Orsk Branch of Samarsk Railway Engineering InstituteOrsk

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