Intersections of valuation overrings of two-dimensional Noetherian domains



We survey and extend recent work on integrally closed overrings of two-dimensional Noetherian domains, where such overrings are viewed as intersections of valuation overrings. Of particular interest are the cases where the domain can be represented uniquely by an irredundant intersection of valuation rings, and when the valuation rings can be chosen from a Noetherian subspace of the Zariski-Riemann space of valuation rings.


Prime Ideal Maximal Ideal Prime Divisor Valuation Ring Noetherian Ring 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abhyankar, S.: On the valuations centered in a local domain. Am. J. Math. 78, 321–348 (1956)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Brewer, J., Mott, J.: Integral domains of finite character. J. Reine Angew. Math. 241, 34–41 (1970)MATHMathSciNetGoogle Scholar
  3. 3.
    Cahen, P.J., Chabert, J.L.: Integer-Valued Polynomials, Mathematical Surveys and Monographs, vol. 48. American Mathematical Society, Providence (1997)Google Scholar
  4. 4.
    Engler, A.J., Prestel, A.: Valued Fields. Springer, Heidelberg (2005)MATHGoogle Scholar
  5. 5.
    Gilmer, R.: Multiplicative ideal theory, Queen’s Papers in Pure and Applied Mathematics, No. 12, Queen’s University, Kingston, Ont. (1968)MATHGoogle Scholar
  6. 6.
    Gilmer, R., Heinzer, W.: Irredundant intersections of valuation rings. Math. Z. 103, 306–317 (1968)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Heinzer, W.: On Krull overrings of a Noetherian domain. Proc. Am. Math. Soc. 22, 217–222 (1969)MATHMathSciNetGoogle Scholar
  8. 8.
    Heinzer, W.: Noetherian intersections of integral domains II. Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), pp. 107–119. Lecture Notes in Math., vol.311, Springer, Berlin (1973)Google Scholar
  9. 9.
    Heinzer, W., Ohm, J.: Noetherian intersections of integral domains. Trans. Am. Math. Soc. 167, 291–308 (1972)MATHMathSciNetGoogle Scholar
  10. 10.
    Heinzer, W., Ohm, J.: Defining families for integral domains of real finite character. Can. J. Math. 24, 1170–1177 (1972)MATHMathSciNetGoogle Scholar
  11. 11.
    Heinzer, W., Rotthaus, C., Sally, J.: Formal fibers and birational extensions. Nagoya Math. J. 131, 1–38 (1993)MATHMathSciNetGoogle Scholar
  12. 12.
    Kwegna-Heubo, O.: Kronecker function rings of transcendental field extensions. Comm. Algebra, to appearGoogle Scholar
  13. 13.
    Loper, K.A., Tartarone, F.: A classification of the integrally closed rings of polynomials containing \(\mathbb{Z}[x]\), J. Commut. Algebra 1, 91–157 (2009)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Nagata, M.: On Krull’s conjecture concerning valuation rings, Nagoya Math. J. 4, 29–33 (1952)MATHMathSciNetGoogle Scholar
  15. 15.
    Nagata, M.: Corrections to my paper “On Krull’s conjecture concerning valuation rings.” Nagoya Math. J. 9, 209–212 (1955)MATHMathSciNetGoogle Scholar
  16. 16.
    Nishimura, J.: Note on Krull domains. J. Math. Kyoto Univ. 15(2), 397–400 (1975)MATHMathSciNetGoogle Scholar
  17. 17.
    Ohm, J.: Some counterexamples related to integral closure in D[[x]]. Trans. Am. Math. Soc. 122, 321–333 (1966)MATHMathSciNetGoogle Scholar
  18. 18.
    Olberding, B.: Irredundant intersections of valuation overrings of two-dimensional Noetherian domains. J. Algebra 318, 834–855 (2007)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Olberding, B.: Overrings of two-dimensional Noetherian domains representable by Noetherian spaces of valuation rings. J. Pure Appl. Algebra 212, 1797–1821 (2008)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Olberding, B.: Holomorphy rings in function fields. In: Multiplicative Ideal Theory in Commutative Algebra, pp. 331–348. Springer, New York (2006)Google Scholar
  21. 21.
    Olberding, B.: Noetherian space of integrally closed domains with an application to intersections of valuation rings, Comm. Algebra, to appearGoogle Scholar
  22. 22.
    Olberding, B.: On Matlis domains and Prüfer sections of Noetherian domains. In: Commutative Algebra and its Applications, pp. 321–332. de Gruyter, Berlin (2009)Google Scholar
  23. 23.
    Zariski, O., Samuel, P.: Commutative algebra. Vol. II. Graduate Texts in Mathematics, Vol. 29. Springer, New York (1975)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNew Mexico State UniversityLas CrucesUSA

Personalised recommendations