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On inert modules over valuation domains

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Abstract

We start the investigation of inert modules over valuation domains, a class of modules containing finitely generated and quasi-injective modules. A complete description is provided when the valuation domain is a DVR. For arbitrary valuation domains, we reduce the investigation to reduced torsion modules and obtain a complete characterization of inert uniserial modules.

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References

  1. W. Brandal, Almost maximal integral domains and finitely generated modules. Trans. Am. Math. Soc. 183, 203–222 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Dikranjan, A. Giordano Bruno, L. Salce, S. Virili, Fully inert subgroups of divisible Abelian groups. J. Group Theory 16(6), 915–939 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Dikranjan, A. Giordano Bruno, L. Salce, S. Virili, Intrinsic algebraic entropy. J. Pure Appl. Algebra 219(7), 2933–2961 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Dikranjan, L. Salce, P. Zanardo, Fully inert subgroups of free Abelian groups. Period. Math. Hung. 69(1), 69–78 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. L. Fuchs, L. Salce, Modules over Valuation Domains (Marcel Dekker, New York, 1985)

    MATH  Google Scholar 

  6. L. Fuchs, L. Salce, Modules Over Non-Noetherian Domains, vol. 84 (AMS, New York, 2001)

    MATH  Google Scholar 

  7. B. Goldsmith, L. Salce, P. Zanardo, Fully inert submodules of torsion-free modules over the ring of \(p\)-adic integers. Colloq. Math. 136(2), 169–178 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. B. Goldsmith, L. Salce, P. Zanardo, Fully inert subgroups of Abelian \(p\)-groups. J. Algebra 419, 332–349 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. I. Kaplansky, Maximal fields with valuations. Duke Math. J. 9, 303–321 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Nishi, On the ring of endomorphisms of an indecomposable injective module over a Prüfer ring. Hiroshima Math. J. 2, 271–283 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Ostrowsky, Untersuchungen zur arithmetischen Theorie der Körper. Math. Z. 39, 269–244 (1935)

    Article  MathSciNet  Google Scholar 

  12. L. Salce, Invariants for quasi-injective modules over valuation domains. Colloq. Math. 144(1), 77–86 (2016)

    MathSciNet  MATH  Google Scholar 

  13. L. Salce, P. Zanardo, Some cardinal invariants for valuation domains. Rend. Sem. Mater. Univ. Padova 74, 205–217 (1985)

    MathSciNet  MATH  Google Scholar 

  14. L. Salce, P. Zanardo, On two-generated modules over valuation domains. Arch. Math. 46, 408–418 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  15. T.S. Shores, W.J. Lewis, Serial modules and endomorphism rings. Duke Math. J. 41, 889–909 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  16. R. Warfield, Decomposition of injective modules. Pac. J. Math. 31, 263–276 (1969)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova, Via Trieste 63, 35121, Padua, Italy

    Luigi Salce

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  1. Luigi Salce
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Correspondence to Luigi Salce.

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Salce, L. On inert modules over valuation domains. Period Math Hung 79, 120–133 (2019). https://doi.org/10.1007/s10998-019-00285-2

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