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Holomorphy rings of function fields

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Multiplicative Ideal Theory in Commutative Algebra

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References

  1. S. Abhyankar. On the valuations centered in a local domain. Amer. J. Math., 78:321–348, 1956.

    Article  MathSciNet  MATH  Google Scholar 

  2. D. D. Anderson and B. G. Kang. Pseudo-Dedekind domains and divisorial ideals in R[X]T. J. Algebra, 122(2):323–336, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  3. C. Andradas. Real places in function fields. Comm. Algebra, 13(5):1151–1169, 1985.

    MathSciNet  MATH  Google Scholar 

  4. J. Bochnak, M. Coste, and M.-F. Roy. Real algebraic geometry, volume 36 of Ergebniase der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1998.

    Google Scholar 

  5. E. Becker. The real holomorphy ring and sums of 2nth powers. In Real algebraic geometry and quadratic forms (Rennes, 1981), volume 959 of Lecture Notes in Math., pages 139–181. Springer, Berlin, 1982.

    Google Scholar 

  6. R. Berr. On real holomorphy rings. In Real analytic and algebraic geometry (Trento, 1992), pages 47–66. de Gruyter, Berhn, 1995.

    Google Scholar 

  7. E. Becker and B. Jacob. Rational points on algebraic varieties over a generalized real closed field: a model theoretic approach. J. Reine Angew. Math., 357:77–95, 1985.

    MathSciNet  MATH  Google Scholar 

  8. M. A. Buchner and W. Kucharz. On relative real holomorphy rings. manuscripta math., 63(3):303–316, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  9. L. Bröcker and H.-W. Schülting. Valuations of function fields from the geometrical point of view. J. Reine Angew. Math., 365:12–32, 1986.

    MathSciNet  MATH  Google Scholar 

  10. A. Dress. Lotschnittebenen mit halbierbarem rechtem Winkel. Arch. Math., 16:388–392, 1965.

    Article  MathSciNet  MATH  Google Scholar 

  11. L. Fuchs, W. Heinzer, and B. Olberding. Commutative ideal theory without finiteness conditions: primal ideals. Trans. Amer. Math. Soc., 357(7):2771–2798, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Fontana, J. A. Huckaba, and I. J. Papick. Prüfer domains, volume 203 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York, 1997.

    Google Scholar 

  13. R. Gilmer. Two constructions of Prüfer domains. J. Reine Angew. Math., 239/240:153–162, 1969.

    MathSciNet  Google Scholar 

  14. R. Gilmer. Multiplicative ideal theory, volume 90 of Queen’s Papers in Pure and Applied Mathematics. Queen’s University, Kingston, ON, 1992.

    Google Scholar 

  15. I. Kaplansky. Commutative rings. The University of Chicago Press, Chicago, Ill.-London, 1974.

    MATH  Google Scholar 

  16. F.-V. Kuhlmann and A. Prestel. On places of algebraic function fields. J. Reine Angew. Math., 353:181–195, 1984.

    MathSciNet  MATH  Google Scholar 

  17. F.-V. Kuhlmann. Places of algebraic function fields in arbitrary characteristic. Adv. Math., 188(2):399–424, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  18. F.-V. Kuhlmann. Value groups, residue fields, and bad places of rational function fields. Trans. Amer. Math. Soc., 356(11):4559–4600 (electronic), 2004.

    Article  MathSciNet  MATH  Google Scholar 

  19. K. A. Loper. On Prüfer non-D-rings. J. Pure Appl. Algebra, 96(3):271–278, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  20. K. A. Loper. Ideals of integer-valued polynomial rings. Comm. Algebra, 25(3):833–845, 1997.

    MathSciNet  MATH  Google Scholar 

  21. S. MacLane and O. F. G. Schilling. Zero-dimensional branches of rank one on algebraic varieties. Ann. of Math. (2), 40:507–520, 1939.

    Google Scholar 

  22. N. Nakano. Idealtheorie in eineni speziellen unendlichen algebraischen Zahlkörper. J. Sci. Hiroshima Univ. Ser. A., 16:425–439, 1953.

    MathSciNet  MATH  Google Scholar 

  23. B. Olberding and M. Roitnian. The minimal number of generators of an invertible ideal, this volume.

    Google Scholar 

  24. P. Ribenboim. Remarks on existentially closed fields and Diophantine equations. Rend. Sem. Mat. Univ. Padova, 71:229–237, 1984.

    MathSciNet  MATH  Google Scholar 

  25. P. Roquette. Principal ideal theorems for holomorphy rings in fields. J. Reine Angew. Math., 262/263:361–374, 1973.

    MathSciNet  Google Scholar 

  26. D. Rush. Bezout domains with stable range 1. J. Pure Appl. Algebra, 158:309–324, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  27. H.-W. Schülting. On real places of a field and their holomorphy ring. Comm. Algebra, 10(12):1239–1284, 1982.

    MathSciNet  MATH  Google Scholar 

  28. H.-W. Schülting. Real holomorphy rings in real algebraic geometry. In Real algebraic geometry and quadratic forms (Rennes, 1981), volume 959 of Lecture Notes in Math., pages 433–442. Springer, Berlin, 1982.

    Google Scholar 

  29. M. Zafrullah. On generalized Dedekind domains. Mathematika, 33(2):285–295 (1987), 1986.

    Article  MathSciNet  Google Scholar 

  30. O. Zariski and P. Samuel. Commutative algebra. Vol. II. Springer-Verlag, New York, 1975.

    MATH  Google Scholar 

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Olberding, B. (2006). Holomorphy rings of function fields. In: Brewer, J.W., Glaz, S., Heinzer, W.J., Olberding, B.M. (eds) Multiplicative Ideal Theory in Commutative Algebra. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-36717-0_20

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