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Countable mixed Abelian groups with very nice full-rank subgroups

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Abstract

We exhibit a maximal set of 2N0 “almost rigid” countable mixed abelian groupsG with the same prescribed torsion subgroupstG, the same quotientG/tG and a fixed countable and cotorsion-free ringA such that EndG/Hom(G, tG)≌A. Despite the fact that these candidates will not allow any structure theorem, they are close relatives of the well-behaving family of Warfield groups. The results are developed in a module category for suitable ground rings.

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References

  1. A. L. S. Corner,Every countable reduced torsion-free ring is an endomorphism ring, Proc. London Math. Soc. (3)13 (1963), 687–710.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. L. S. Corner and R. Göbel,Prescribing endomorphism algebras — a unified treatment, Proc. London Math. Soc.50 (1985).

  3. A. L. S. Corner,Every countable reduced torsion-free algebra is an endomorphism algebra, unpubl. manuscript.

  4. M. Dugas,On the existence of large mixed modules, inAbelian Group Theory, Proceedings Honolulu 1982/3, Springer LNM1006 (1983), 412–424.

    MathSciNet  Google Scholar 

  5. M. Dugas and R. Göbel,On endomorphism rings of primary abelian groups, Math. Ann.261 (1982), 359–385.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Dugas and R. Göbel,Every cotorsion-free algebra is an endomorphism algebra, Math. Z.181 (1982), 451–471.

    Article  MATH  MathSciNet  Google Scholar 

  7. L. Fuchs,Infinite Abelian Groups, Vol. I (1970), Vol. II (1974), Academic Press, New York.

  8. L. Fuchs,Abelian p-groups and mixed groups, Montreal Lecture Notes, Les Presses de l’Université, 1980.

  9. I. Kaplansky and G. Mackey,A generalization of Ulm’s theorem, Summa Brasil. Math.2 (1951), 195–202.

    MathSciNet  Google Scholar 

  10. R. Kuhl-Prelle,Endomorphismen-Algebren von Moduln, Staatsexamens-thesis, Universität Essen (1978).

  11. W. May and E. Toubassi,Endomorphisms of abelian groups and the theorem of Baer and Kaplansky, J. Algebra43 (1976), 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  12. C. Megibben,Modules over an incomplete discrete valuation ring, Proc. Am. Math. Soc.19 (1968), 450–452.

    Article  MATH  MathSciNet  Google Scholar 

  13. F. Richman,Mixed local groups, inAbelian Group Theory, Proceedings Oberwolfach 1981, Springer LNM874 (1981), 374–404.

    MathSciNet  Google Scholar 

  14. F. Richman,Mixed abelian groups, inAbelian Group Theory, Proceedings Honolulu 1982/83, Springer LNM1006 (1983), 445–470.

    MathSciNet  Google Scholar 

  15. R. B. Warfield,The structure of mixed abelian groups, inAbelian Group Theory, Proceedings of the 2nd New Mexico State University Conference 1976, Springer LNM616 (1976), 1–38.

    MathSciNet  Google Scholar 

  16. R. B. Warfield,Classification theory of abelian groups, II: Local theory, inAbelian Group Theory, Proceedings Oberwolfach 1981, Springer LNMbd874 (1981), 322–349.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematik, Universität Essen, GHS, Fachbereich 6, D 43, Essen, West Germany

    Manfred Dugas

  2. Institute of Mathematics and Computer Science, The Hebrew University of Jerusalem, Givat Ram, 91904, Jerusalem, Israel

    Rüdiger Göbel

Authors
  1. Manfred Dugas
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  2. Rüdiger Göbel
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Dugas, M., Göbel, R. Countable mixed Abelian groups with very nice full-rank subgroups. Israel J. Math. 51, 1–12 (1985). https://doi.org/10.1007/BF02772953

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  • DOI: https://doi.org/10.1007/BF02772953

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