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Arabian Journal of Mathematics

, Volume 7, Issue 4, pp 249–271 | Cite as

Pointwise minimal extensions

  • Paul-Jean Cahen
  • Gabriel Picavet
  • Martine Picavet-L’Hermitte
Open Access
Article

Abstract

We characterize pointwise minimal extensions of rings, introduced by Cahen et al. (Rocky Mt J Math 41:1081–1125, 2011), in the special context of domains. We show that pointwise minimal extensions are either integral or integrally closed. In the closed case, they are nothing but minimal extensions. Otherwise, there are four cases: either all minimal sub-extensions are of the same type (ramified, decomposed, or inert) or coexist as only ramified and inert minimal sub-extensions.

Mathematics Subject Classification

13B02 13B21 13B22 13B30 

Notes

References

  1. 1.
    Bourbaki, N.: Algèbre, Chs 4–7. Masson, Paris (1981)zbMATHGoogle Scholar
  2. 2.
    Cahen, P.-J.: Couples d’anneaux partageant un idéal. Archiv der Math. 51, 505–514 (1988)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cahen, P.-J.; Dobbs, D.E.; Lucas, T.G.: Characterizing minimal ring extensions. Rocky Mt. J. Math. 41(4), 1081–1125 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cahen, P.-J.; Dobbs, D.E.; Lucas, T.G.: Valuative domains. J. Algebra Appl. 9, 43–72 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dobbs, D.E.; Picavet, G.; Picavet-L’Hermitte, M.: Characterizing the ring extensions that satisfy FIP or FCP. J. Algebra 371, 391–429 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dobbs, D.E.; Picavet, G.; Picavet-L’Hermitte, M.: Transfer results for the FIP and FCP properties of ring extensions. Commun. Algebra 43, 1279–1316 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dobbs, D.E.; Shapiro, J.: Patching together a minimal overring. Houston J. Math. 36, 985–995 (2010)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ferrand, D.; Olivier, J.-P.: Homomorphismes minimaux d’anneaux. J. Algebra 16, 461–471 (1970)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Grothendieck, A.; Dieudonné, J.A.: Eléments de Géométrie Algébrique I. Springer, Berlin (1971)zbMATHGoogle Scholar
  10. 10.
    Picavet, G.; Picavet-L’Hermitte, M.: Morphismes t-clos. Commun. Algebra 21, 179–219 (1993)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Picavet, G.; Picavet-L’Hermitte, M.: T-closedness. In: Chapman, S.T., Glaz, S. (eds.) Non-Noetherian Commutative Ring Theory, Mathematics and Its Applications, vol. 520, pp. 369–386. Springer, New York (2000)CrossRefGoogle Scholar
  12. 12.
    Picavet, G.; Picavet-L’Hermitte, M.: Some more combinatorics results on Nagata extensions. Palest. J. Math. 5, 49–62 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Picavet-L’Hermitte, M.: Minimal order morphisms. J. Number Theory 45, 1–27 (1993)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Swan, R.G.: On seminormality. J. Algebra 67, 210–229 (1980)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Traverso, C.: Seminormality and Picard group. Ann. Scuola Norm. Sup. Pisa 24, 585–595 (1970)MathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Aix en ProvenceFrance
  2. 2.MathématiquesLe CendreFrance

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