Constructing Examples of Integral Domains by Intersecting Valuation Domains

  • K. Alan Loper
Part of the Mathematics and Its Applications book series (MAIA, volume 520)


The subject of this survey is a method of constructing integral domains which is not often utilized compared to various other methods, but is deceptively powerful. The motivation is the following classical theorem of W. Krull.


Prime Ideal Maximal Ideal Integral Domain Principal Ideal Residue Field 
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  1. 1.
    Arnold, J. and Sheldon, P., Integral domains that satisfy Gauss’s Lemma, Michigan Math. J. 22, 39–51 (1975).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Cahen, P.-J. and Chabert, J.-L., Integer-valued polynomials, Mathematical Surveys and Monographs No. 48, American Mathmatical Society, Providence, RI, 1997. CMP 97:04zbMATHGoogle Scholar
  3. 3.
    Chabert, J.-L., Un anneau de Prüfer, J. Algebra. 107, 1–17 (1987).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chabert, J.-L., Anneaux de Pôlynomes à valeurs entières et anneaux de Prüfer, C. R. Acad. Sci. Paris. 312, 715–720 (1991).MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chabert, J.-L., Integer-valued polynomials, Prüfer domains, and localization, Proc. Amer. Math. Soc. 118, 1061–1073 (1993).MathSciNetzbMATHGoogle Scholar
  6. 6.
    Dress, A., Lotschnittebenen mit halbierbarem rechten Winkel, Arch. Math. 16, 388–392,(1965).MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Fontana, M. and Huckaba, J. and Papick, I., Prüfer domains, Marcel Dekker, New York, 1997. 95–100 (1989).zbMATHGoogle Scholar
  8. 8.
    Fontana, M. and Gabelli, S., Prüfer domains with class group generated by the classes of the invertible maximal ideals, Communications in Algebra, 25, 3993–4008 (1997). 95–100 (1989).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Gilmer, R., Two constructions of Prüfer domains, J. reine angew. Math. 239, 153–162 (1970).MathSciNetGoogle Scholar
  10. 10.
    Gilmer, R., Prüfer domains and rings of integer-valued polynomials, J. Algebra. 129, 502–517 (1990).MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Gunji, H. and McQuillan, D, On rings with a certain divisibility proberty, Michigan Math. J. 22, 289–299 (1975).MathSciNetzbMATHGoogle Scholar
  12. 12.
    Heitmann, R., Generating ideals in Prüfer domains, Pacific J. Math. 62, 117–126 (1976).MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Henriksen, M., On the prime ideals of the ring of entire functions, Pacific J. Math. 3, 711–720 (1953).MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kucharz, W., Invertible ideals in real holomorphy rings, J. reine angew. Math. 395, 171–185 (1989).MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kucharz, W., Generating ideals in real holomorphy rings, J. Algebra. 144, 1–7 (1991).MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Loper, K.A., On Prüfer non-D-rings, J. Pure Appl. Algebra, 96, 271–278 (1994).MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Loper, K.A., More almost Dedekind domains and Prüfer domains of polynomials in Zero-dimensional Commutative Rings, D.F. Anderson and D. E. Dobbs, editors, Marcel Dekker, New York, (1995).Google Scholar
  18. 18.
    Loper, K.A., Sequence domains and integer-valued polynomials, J. Pure Appl. Algebra, 119, 185–210 (1997).MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Loper, K.A., A class of Prüfer domains that are similar to the ring of entire functions, Rocky Mountain J. of Math, 28, 267–285 (1998).MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Loper, K. A., Two Prüfer domain counterexamples, to appear in the Journal of Algebra.Google Scholar
  21. 21.
    McQuillan, D., On Prüfer domains of polynomials, J. reine angew. Math. 358, 162–178 (1985).MathSciNetzbMATHGoogle Scholar
  22. 22.
    Swan, R., n-generator ideals in Prüfer domains, Pacific J. Math. 111, 433–446 (1984).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • K. Alan Loper
    • 1
  1. 1.Department of MathematicsOhio State University — NewarkNewarkUSA

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