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On divisible modules over domains

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Abelian Groups and Modules

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 287))

Abstract

This note is devoted to the study of divisible modules over arbitrary commutative domains R with 1. Whereas divisible modules over Dedekind domains can be completely characterized by numerical invariants, not much is known about their structures in the general case. Divisible modules have been studied by Matlis [6] who was the first to distinguish between divisibility in general and h-divisibility. He established an important duality between h-divisible torsion modules and complete torsion-free R-modules [7] . Matlis [6] and Hamsher [4] characterized those domains R for which all divisible R-modules are h-divisible, e.g. by the property that the field Q of quotients of R has projective dimension 1.

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References

  1. Auslander, L., On the dimension of modules and algebras, III, Nagoya Math. J. 9 (1955), 67–77.

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  2. de la Rosa, B. and Fuchs, L., On h-divisible torsion modules over domains, to appear.

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  3. Fuchs, L., On projective dimensions of modules over valuation domains, Abelian Group Theory, Lecture Notes in Math., Springer, 1006 (1983), 589–598.

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  4. Hamsher, R.M., On the structure of a one-dimensional quotient field, Journ. of Alg. 19 (1971), 416–425.

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  5. Kaplansky, I., The homological dimension of a quotient field, Nagoya Math. J. 27 (1966), 139–142.

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  6. Matlis, E., Divisible modules, Proc. Amer. Math. Soc. 11 (1960), 385–391.

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Author information

Authors and Affiliations

  1. Tulane University, New Orleans, La., USA

    L. Fuchs

Authors
  1. L. Fuchs
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Editor information

Editors and Affiliations

  1. Universität Essen, Germany

    R. Göbel

  2. Universita’ di Padova, Italy

    C. Metelli  & A. Orsatti  & 

  3. Universita’ di Udine, Italy

    L. Salce

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© 1984 Springer-Verlag Wien

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Fuchs, L. (1984). On divisible modules over domains. In: Göbel, R., Metelli, C., Orsatti, A., Salce, L. (eds) Abelian Groups and Modules. International Centre for Mechanical Sciences, vol 287. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2814-5_25

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  • DOI: https://doi.org/10.1007/978-3-7091-2814-5_25

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-81847-3

  • Online ISBN: 978-3-7091-2814-5

  • eBook Packages: Springer Book Archive

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