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References

  1. D.D. Anderson, Some remarks on the ring R(X), Comment. Math. Univ. St. Pauli XXVI-2 (1977), 137–140.

    Google Scholar 

  2. D.D. Anderson and D.F. Anderson, Generalized GCD domains, Comment. Math. Univ. St. Pauli 28 (1979), 215–221.

    Google Scholar 

  3. D.D. Anderson, D. Dumitrescu, and M. Zafrullah, Quasi-Schreier domains II, manuscript.

    Google Scholar 

  4. D.D. Anderson and D. Kwak, The u,u −1-lemma revisited, Comm. Algebra 24 (1996), 2447–2454.

    MathSciNet  Google Scholar 

  5. D.D. Anderson and M. Zafrullah, On t-invertihility III, Comm. Algebra 21 (1993), 1189–1201.

    MathSciNet  Google Scholar 

  6. A. Ayache, P.-J. Cahen, and O. Echi, Anneaux quasi-prüjeriens et P-anneaux, Boll. Un. Mat. Ital. B(7) 10 (1996), 1–24.

    MathSciNet  Google Scholar 

  7. J. Arnold and R. Gilmer, On the contents of polynomials, Proc. Amer. Math. Soc. 24 (1970), 556–562.

    Article  MathSciNet  Google Scholar 

  8. J. Arnold and R. Gilmer, The dimension sequence of a commutative ring, Amer. J. Math. 96 (1974), 385–408.

    Article  MathSciNet  Google Scholar 

  9. J. Arnold and R. Gilmer, Two questions concerning dimension sequences. Arch. Math. (Basel) 29 (1977), 497–503.

    MathSciNet  Google Scholar 

  10. J. Brewer and W. Heinzer, Associated primes of principal ideals, Duke Math. J. 41 (1974), 1–7.

    Article  MathSciNet  Google Scholar 

  11. P.-J. Cahen, E. Houston, and T. Lucas, Discrete valuation overrings of Noetherian domains, Proc. Amer. Math. Soc. 124 (1996), 1719–1721.

    Article  MathSciNet  Google Scholar 

  12. C. Chevalley, La notion d’anneau de décomposition, Nagoya Math. J. 7 (1954), 21–33.

    MathSciNet  Google Scholar 

  13. J. Dieudonné, Sur la théorie de la divisibilité. Bull. Soc. Math. France 69 (1941), 133–144.

    MathSciNet  Google Scholar 

  14. A. de Souza Doering and Y. Lequain, Chains of prime ideals in polynomial rings, J. Algebra 78 (1982), 163–180.

    Article  MathSciNet  Google Scholar 

  15. D. Dobbs, On INC-extensions and polynomials with unit content, Canad. Math. Bull. 23 (1980), 37–42.

    MathSciNet  Google Scholar 

  16. D. Dobbs, E. Houston, T. Lucas, M. Roitman, and M. Zafrullah, On t-linked overrings, Comm. Algebra 20 (1992), 1463–1488.

    MathSciNet  Google Scholar 

  17. H. Flanders, A remark on Kronecker’s theorem on forms, Proc. Amer. Math. Soc. 3 (1952), 197.

    Article  MathSciNet  Google Scholar 

  18. M. Fontana and S. Gabelli, On the class group and the local class group of a pullback, J. Algebra 181 (1996), 803–835.

    Article  MathSciNet  Google Scholar 

  19. M. Fontana, S. Gabelli, and E. Houston, UMT-domains and domains with Prüfer integral closure, Comm. Algebra 26 (1998), 1017–1039.

    MathSciNet  Google Scholar 

  20. S. Gabelli and M. Roitman, Maximal divisorial ideals and t-maximal ideals, JP J. Algebra Number Theory Appl. 4 (2004), 323–336.

    MathSciNet  Google Scholar 

  21. R. Gilmer, Some applications of the Hilfssatz von Dedekind-Mertens, Math. Scan. 20 (1967), 240–244.

    MathSciNet  Google Scholar 

  22. R. Gilmer, Multiplicative ideal theory. Queen’s Papers on Pure and Applied Mathematics, No. 12, Queen’s University, Kingston, Ontario, 1968.

    Google Scholar 

  23. R. Gilmer, Multiplicative ideal theory, Dekker, New York, 1972.

    MATH  Google Scholar 

  24. R. Gilmer and J. Hoffmann, A characterization of Prüfer domains in terms of polynomials, Pac. Math. J. 60 (1975), 81–85.

    MathSciNet  Google Scholar 

  25. S. Glaz, Finiteness and differential properties of ideals, Ph.D. thesis, Rutgers University, 1977.

    Google Scholar 

  26. S. Glaz, Commutative coherent rings, Lecture Notes in Mathematics 1371, Springer, Berlin, 1989.

    Google Scholar 

  27. S. Glaz and W. Vasconcelos, Flat ideals III, Comm. Algebra 12 (1984), 199–227.

    MathSciNet  Google Scholar 

  28. M. Griffin, Some results on Prüfer v-multiplication rings, Canad. J. Math. 19 (1967), 710–722.

    MathSciNet  Google Scholar 

  29. E. Hamann, E. Houston, and J. Johnson, Properties of uppers to zero in R[x], Pac. J. Math. 135 (1988), 65–79.

    MathSciNet  Google Scholar 

  30. J. Hedstroni and E. Houston, Pseudo-valuation domains, Pac. J. Math. 75 (1978), 137–147.

    Google Scholar 

  31. J. Hedstroni and E. Houston, Pseudo-valuation domains (II), Houston J. Math. 4 (1978), 199–207.

    MathSciNet  Google Scholar 

  32. W. Heinzer and J. Ohm, An essential ring which is not a v-multiplication ring, Canad. J. Math. 25 (1973), 856–861.

    MathSciNet  Google Scholar 

  33. E. Houston, Prime t-ideals in R[X], in Commutative ring theory. P.-J. Cahen, D. Costa, M. Fontana, S. Kabbaj, eds., Lecture Notes in Pure and Applied Mathematics 153, Dekker, New York, 1994, pp. 163–170.

    Google Scholar 

  34. E. Houston, S. Malik, and J. Mott, Characterizations of *-multiplication domains, Canad. Math. Bull. 27 (1984), 48–52.

    MathSciNet  Google Scholar 

  35. E. Houston and M. Zafrullah, On t-invertibility II, Comm. Algebra 17 (1989), 1955–1969.

    MathSciNet  Google Scholar 

  36. E. Houston and M. Zafrullah, UMV-domains, in Arithmetical properties of commutative rings and modules, S. Chapman, ed., Lecture Notes in Pure and Applied Mathematics 241, Chapman & Hall/CRC, London, 2005, pp. 304–315.

    Google Scholar 

  37. B.C. Kang, Prüfer v-multiplication domains and the ring \( R[X]_{N_v }\), J. Algebra 123 (1989), 151–170.

    Article  MathSciNet  Google Scholar 

  38. I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, 1970.

    MATH  Google Scholar 

  39. W. Krull, Beiträge zur Arithmetik kommutativer Integritätsbereiche II, Math. Z. 41 (1936), 665–679.

    Article  MathSciNet  Google Scholar 

  40. S. Malik and J. Mott, Strong S-domains, J. Pure. Appl. Algebra textbf28 (1983), 249–264.

    Article  MathSciNet  Google Scholar 

  41. S. McAdam, A Noetherian example, Comm. Algebra 4 (1976), 245–247.

    MathSciNet  Google Scholar 

  42. J. Mott and M. Zafrullah, On Prüfer v-multiplication domains, Manuscripta Math. 35 (1981), 1–26.

    Article  MathSciNet  Google Scholar 

  43. J. Mott, B. Nashier, and M. Zafrullah, Contents of polynomials and invertibility, Comm. Algebra 18 (1990), 1569–1583.

    MathSciNet  Google Scholar 

  44. M. Nagata, On Krull’s conjecture concerning valuation rings, Nagoya Math. J. 4 (1952), 29–33.

    MathSciNet  Google Scholar 

  45. M. Nagata, Corrections to my paperOn Krul’’s conjecture concerning valuation rings”, Nagoya Math. J. 9 (1955), 209–212.

    MathSciNet  Google Scholar 

  46. A. Ouertani, Exemples de dimension de Krull d’anneaux de polynômes, Canad. Math. Bull. 33 (1990), 135–138.

    MathSciNet  Google Scholar 

  47. J. Querre, Idéaux divisoriels d’un anneau de polynômes, J. Algebra 64 (1980), 270–284.

    Article  MathSciNet  Google Scholar 

  48. I. Papick, Topologically defined classes of going-down domains, Trans. Anier. Math. Soc. 219 (1976), 1–37.

    Article  MathSciNet  Google Scholar 

  49. I. Papick, A note on Prüfer v-multiplication domains. Boll. Un. Mat. Ital. A(6) 1 (1982), 133–136.

    MathSciNet  Google Scholar 

  50. I. Papick, Super-primitive elements, Pac. J. Math. 105 (1983), 217–226.

    MathSciNet  Google Scholar 

  51. A. Seidenberg, On the dimension theory of rings, Pac. J. Math 3 (1953), 505–512.

    MathSciNet  Google Scholar 

  52. A. Seidenberg, On the dimension theory of rings II, Pac. J. Math. 4 (1954), 603–614.

    MathSciNet  Google Scholar 

  53. H. Tang, Gauss’ lemma, Proc. Amer. Math. Soc. 35 (1972), 372–376.

    Article  MathSciNet  Google Scholar 

  54. M. Zafrullah, On finite conductor domains, Manuscripta Math. 24 (1978), 191–204.

    Article  MathSciNet  Google Scholar 

  55. M. Zafrullah, Various facets of rings between D[X] and K[X], Comm. Algebra 31 (2003), 2497–2540.

    Article  MathSciNet  Google Scholar 

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Houston, E. (2006). Uppers to zero in polynomial rings. 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_15

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