Skip to main content
SpringerLink
Log in

Some questions concerning proper subrings

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

Given a ring extension \(R\subset S\) of integral domains, it is shown that if each proper subring of \(S\) containing \(R\) is integrally closed, then \(S\) is integrally closed. As an application, we show that if each proper subring of \(S\) containing \(R\) is a valuation (resp., Prüfer, resp. Principal ideal) domain, then so is \(S\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ayache, A., Ben Nasr, M., Jarboui, N.: PID pairs of rings and maximal non-PID subrings. Math. Z. 268(3–4), 635–647 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ayache, A., Dobbs, D.E., Echi, O.: On maximal non-ACCP subrings. J. Algebra Appl. 5(6), 873–894 (2007)

    Article  MathSciNet  Google Scholar 

  3. Ayache, A., Jaballah, A.: Residually algebraic pairs of rings. Math. Z. 225, 49–65 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ayache, A., Jarboui, N.: Maximal non-Noetherian subrings of a domain. J. Algebra 2(248), 806–823 (2002)

    Article  MathSciNet  Google Scholar 

  5. Nasr, Ben: M., Jarboui, N.: Maximal non-Jaffard subrings of a field. Publ. Math. 1(44), 157–175 (2000)

    Article  Google Scholar 

  6. Nasr, Ben: M., Jarboui, N.: On maximal non-valuation subrings. Houston J. Math. 1(37), 47–59 (2011)

    Google Scholar 

  7. Davis, E.D.: Overrings of commutative rings III: normal pairs. Trans. Am. Math. Soc. 182, 175–185 (1973)

    MATH  Google Scholar 

  8. Davis, E.D.: Integrally closed pairs. In: Conference on Commutative Algebra Lecture Notes in Mathematics, vol. 311, pp. 103–106 (1973)

  9. Fontana, M.: Topologically defined classes of commutative rings. Ann. Math. Pura Appl. 123, 331–355 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gilmer, R.: Multiplicative Ideal Theory. Dekker, New York (1972)

    MATH  Google Scholar 

  11. Jaballah, A.: Maximal non-Prüfer and maximal non-integrally closed subrings of a field. J. Algebra Appl. 11(5), 18 (2012)

    Article  MathSciNet  Google Scholar 

  12. Jarboui, N.: When is each proper overring of \(R\) an S(eidenberg)-domain? Publ. Math. 2(46), 435–440 (2002)

    Article  MathSciNet  Google Scholar 

  13. Visweswaran, S.: Intermediate rings between \(D+I\) and \(K[y_1,\cdots, y_t]\). Commun. Algebra 2(18), 309–345 (1990)

    Article  MathSciNet  Google Scholar 

  14. Wadsworth, A.R.: Pairs of domains where all intermediate domains are Noetherian. Trans. Amer. Math. Soc. 195, 201–211 (1974)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors extend their thanks to the referee for his/her valuable suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, P.O. Box: 1171, 3000 , Sfax, Tunisia

    Noômen Jarboui, Manar El Islam Toumi & Salma Trabelsi

Authors
  1. Noômen Jarboui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manar El Islam Toumi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Salma Trabelsi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Noômen Jarboui.

Additional information

Communicated by Marco Fontana.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jarboui, N., Toumi, M.E.I. & Trabelsi, S. Some questions concerning proper subrings. Ricerche mat. 64, 51–55 (2015). https://doi.org/10.1007/s11587-014-0188-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-014-0188-6

Keywords

Mathematics Subject Classification (2010)

Search

Navigation