Recent Progress on Going-Down I

  • David E. Dobbs
Part of the Mathematics and Its Applications book series (MAIA, volume 520)

Abstract

The years 1970–77 witnessed considerable research activity in connection with the “going-down” concept. To chronicle that activity and the sub­ject’s earlier history, Ira Papick and I wrote a survey [78] which appeared in 1978. Since then, work in this area has continued unabated, and I propose to survey most of the post-1977 work concerning “going-down.” Because of limitations of space, our focus here is almost exclusively on papers of which I was either the author or a coauthor. In doing so, I take this opportunity to thank my 24 coauthors whose work is referenced here, while begging the indulgence of the few authors whose post-1977 work on “going-down” goes unmentioned here.

Keywords

Prime Ideal Integral Domain Zariski Topology Ring Extension Valuation Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. D. Anderson, D. F. Anderson, D. L. Costa, D. E. Dobbs, J. L. Mott and M. Zafrullah, Some characterizations of v-domains and related properties, Colloq. Math., 58 (1989), 1–9.MathSciNetMATHGoogle Scholar
  2. [2]
    D. D. Anderson, D. F. Anderson, D. E. Dobbs and E. G. Houston, Some finiteness and divisibility conditions on the proper overrings of an integral domain, Comm. Algebra, 12 (1984), 1689–1706.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    D. D. Anderson, D. E. Dobbs and B. Mullins, The primitive element theorem for commu­tative algebras, Houston J. Math., 25 (1999), 603–623.MathSciNetGoogle Scholar
  4. [4]
    D. F. Anderson, A. Badawi and D. E. Dobbs, Pseudo-valuation rings, in Commutative Ring Theory, II, Lecture Notes Pure Appl. Math., 185, Dekker, New York (1997), pp. 57–67.Google Scholar
  5. [5]
    D. F. Anderson, A. Badawi and D. E. Dobbs, Pseudo-valuation rings, II, Boll. Un. Mat. Ital., to appear.Google Scholar
  6. [6]
    D. F. Anderson, V. Barucci and D. E. Dobbs, Coherent Mori domains and the principal ideal theorem, Comm. Algebra, 15 (1987), 1119–1156.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    D. F. Anderson, A. Bouvier, D.E. Dobbs, M. Font ana and S. Kabbaj, On Jaffard domains, Exposition. Math., 6 (1988), 145–175.MathSciNetMATHGoogle Scholar
  8. [8]
    D. F. Anderson and D. E. Dobbs, Pairs of rings with the same prime ideals, Canad. J. Math., 32 (1980), 362–384.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    D. F. Anderson, D. E. Dobbs and M. Fontana, On treed Nagata rings, J. Pure Appl. Algebra, 61 (1989), 107–122.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    D. F. Anderson, D. E. Dobbs and M. Fontana, Notes on N-dimension sequences, Math. Japonica, 36 (1991), 121–125.MathSciNetMATHGoogle Scholar
  11. [11]
    D. F. Anderson, D. E. Dobbs, M. Fontana and M. Khalis, Catenarity of formal power series rings over a pullback, J. Pure Appl. Algebra, 78 (1992), 109–122.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    D. F. Anderson, D. E. Dobbs and J. A. Huckaba, On seminormal overrings, Comm. Al­gebra, 10 (1982), 1421–1448.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    D. F. Anderson, D. E. Dobbs, S. Kabbaj and S. B. Mulay, Universally catenarian domains of D + M type, Proc. Amer. Math. Soc, 104 (1988), 378–384.MathSciNetMATHGoogle Scholar
  14. [14]
    D. F. Anderson, D. E. Dobbs and M. Roitman, Root closure in commutative rings, Ann. Sei. Univ. Clermont II, Sér. Math. 26 (1990), 1–11.MathSciNetGoogle Scholar
  15. [15]
    A. Andreotti and E. Bombieri, Sugli omeomorfismi délie varietà algebriche, Ann. Scuola Norm. Sup. Pisa, 23 (1969), 431–450.MathSciNetMATHGoogle Scholar
  16. [16]
    J. T. Arnold and R. Gilmer, The dimension sequence of a commutative ring, Amer. J. Math., 96 (1974), 385–408.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, 1969.MATHGoogle Scholar
  18. [18]
    A. Ayache, Quelques remarques sur les conditions de chaines, Arch, der Math., 72 (1999), 270–277.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    A. Badawi and D. E. Dobbs, On locally divided rings and going-down rings, submitted for publication.Google Scholar
  20. [20]
    V. Barucci, D. E. Dobbs and M. Fontana, Conducive integral domains as pullbacks, Manuscripta Math., 54 (1986), 261–277.MathSciNetCrossRefGoogle Scholar
  21. [21]
    V. Barucci, D. E. Dobbs and M. Fontana, Gorenstein conducive domains, Comm. Algebra, 18 (1990), 3889–3903.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    V. Barucci, D. E. Dobbs and S. B. Mulay, Integrally closed factor domains, Bull. Austral. Math. Soc, 37 (1988), 353–366.MathSciNetCrossRefGoogle Scholar
  23. [23]
    E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form D + M, Michigan Math. J., 20 (1973), 79–95.MathSciNetMATHGoogle Scholar
  24. [24]
    M. B. Boisen, Jr. and P. B. Sheldon, CPI-extensions: overrings of domains with special prime spectrums, Canad. J. Math., 29 (1977), 722–737.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    A. Bouvier, D. E. Dobbs and M. Fontana, Universally catenarian integral domains, Ad­vances in Math., 72 (1988), 211–238.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    A. Bouvier, D. E. Dobbs and M. Fontana, Two sufficient conditions for universal catenar­ity, Comm. Algebra, 15 (1987), 861–872.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    P.-J. Cahen, J.-L. Chabert, D. E. Dobbs and F. Tartarone, On locally divided domains of the form Int(D), Arch. Math. 74(2000), 183–191.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    R. D. Chatham, Going down pairs of commutative rings, Ph. D. dissertation, Univ. Ten­nessee, Knoxville, 2000.Google Scholar
  29. [29]
    J. T. Condo, J. Coykendall and D. E. Dobbs, Formal power series rings over zero-dimensional SFT-rings, Comm. Algebra, 24 (1996), 2687–2698.MathSciNetMATHGoogle Scholar
  30. [30]
    J. Coykendall and D. E. Dobbs, Fragmented domains have infinite Krull dimension, sub­mitted for publication.Google Scholar
  31. [31]
    J. Coykendall, D. E. Dobbs and B. Mullins, On integral domains with no atoms, Comm. Algebra, 27 (1999), 5813–5831.MathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    D. E. Dobbs, Divided rings and going-down, Pac. J. Math., 67 (1976), 353–363.MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    D. E. Dobbs, On locally divided integral domains and CPI-overrings, Internat. J. Math. & Math. Sci., 4 (1981), 119–135.MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    D. E. Dobbs, Ahmes expansions of formal Laurent series and a class of Non-Archimedean integral domains, J. Algebra, 103 (1986), 193–201.MathSciNetCrossRefMATHGoogle Scholar
  35. [35]
    D. E. Dobbs, Coherent pseudo-valuation domains have Noetherian completions, Houston J. Math., 14 (1988), 481–486.MathSciNetMATHGoogle Scholar
  36. [36]
    D. E. Dobbs, Using integral closure to characterize going-down domains, C. R. Math. Rep. Acad. Sei. Canada, 9 (1987), 155–160.MathSciNetMATHGoogle Scholar
  37. [37]
    D. E. Dobbs, Going-down underrings, Bull. Austral. Math. Soc, 36 (1987), 503–513.MathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    D. E. Dobbs, A note on universally going-down, Math. J. Okayama U., 30 (1988), 1–4.MathSciNetMATHGoogle Scholar
  39. [39]
    D. E. Dobbs, On treed overrings and going-down domains, Rend. Mat., 7 (1987), 317–322.MathSciNetMATHGoogle Scholar
  40. [40]
    D. E. Dobbs, On universally going-down underrings, Arch. Math., 54 (1990), 260–262.MathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    D. E. Dobbs, On universally going-down underrings, II, Arch. Math., 55 (1990), 135–138.MathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    D. E. Dobbs, Rings of formal power series with homeomorphic prime spectra, Rend Circ. Mat. Palermo, Serie II, 41 (1992), 55–61.MathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    D. E. Dobbs, A note on strong locally divided domains, Tsukuba J. Math., 15 (1991), 215–217.MathSciNetMATHGoogle Scholar
  44. [44]
    D. E. Dobbs, Prüfer’s ascent result using INC, Comm. Algebra, 23 (1995), 5413–5417.MathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    D. E. Dobbs, Locally Henselian going-down domains, Comm. Algebra, 24 (1996), 1621–1635.MathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    D. E. Dobbs, Absolutely infective integral domains, Houston J. Math., 22 (1996), 485–497.MathSciNetMATHGoogle Scholar
  47. [47]
    D. E. Dobbs, On characterizations of Prüfer domains using polynomials with unit content, in Factorization in Integral Domains, Lecture Notes Pure Appl. Math., 189, Dekker, New York (1997), pp. 295–303.MathSciNetGoogle Scholar
  48. [48]
    D. E. Dobbs, On Henselian pullbacks, in Factorization in Integral Domains, Lecture Notes Pure Appl. Math., 189, Dekker, New York (1997), pp. 317–326.MathSciNetGoogle Scholar
  49. [49]
    D. E. Dobbs, On flat divided prime ideals, in Factorization in Integral Domains, Lecture Notes Pure Appl. Math., 189, Dekker, New York (1997), pp. 305–315.MathSciNetGoogle Scholar
  50. [50]
    D. E. Dobbs, Going-down rings with zero-divisors, Houston J. Math., 23 (1997), 1–12.MathSciNetMATHGoogle Scholar
  51. [51]
    D. E. Dobbs, A going-up theorem for arbitrary chains of prime ideals, Comm. Algebra, 27 (1999), 3887–3894.MathSciNetCrossRefMATHGoogle Scholar
  52. [52]
    D. E. Dobbs, Lifting chains of prime ideals to paravaluation rings, Rend. Circ. Mat. Palermo, to appear.Google Scholar
  53. [53]
    D. E. Dobbs, A conductor theorem, Houston J. Math., to appear.Google Scholar
  54. [54]
    D. E. Dobbs, Extensions of commutative rings in which trees of prime ideals contract to trees, submitted for publication.Google Scholar
  55. [55]
    D. E. Dobbs, Integral domains with almost integral proper overrings, submitted for publi­cation.Google Scholar
  56. [56]
    D. E. Dobbs and R. Fedder, Conducive integral domains, J. Algebra, 86 (1984), 494–510.MathSciNetCrossRefMATHGoogle Scholar
  57. [57]
    D. E. Dobbs, R. Fedder and M. Fontana, Abstract Riemann surfaces of integral domains and spectral spaces, Ann. Mat. Pura Appl., 148 (1987), 101–115.MathSciNetCrossRefMATHGoogle Scholar
  58. [58]
    D. E. Dobbs, R. Fedder and M. Fontana, G-domains and spectral spaces, J. Pure Appl. Algebra, 51 (1988), 89–110.MathSciNetCrossRefMATHGoogle Scholar
  59. [59]
    D. E. Dobbs and M. Fontana, Classes of commutative rings characterized by their going-up and going-down behavior, Rend, Sem. Mat. Univ. Padova, 66 (1982), 113–127.MathSciNetMATHGoogle Scholar
  60. [60]
    D. E. Dobbs and M. Fontana, Locally pseudo-valuation domains, Ann. Mat. Pura Appl., 134 (1983), 147–168.MathSciNetCrossRefMATHGoogle Scholar
  61. [61]
    D. E. Dobbs and M. Font ana, On pseudo-valuation domains and their globalizations, in Proceedings of Trento Conference, Lecture Notes Pure Appl. Math., 84, Dekker, New York (1983), pp. 65–77.Google Scholar
  62. [62]
    D. E. Dobbs and M. Fontana, Universally going-down homomorphisms of commutative rings, J. Algebra, 90 (1984), 410–429.MathSciNetCrossRefMATHGoogle Scholar
  63. [63]
    D. E. Dobbs and M. Fontana, Universally going-down integral domains, Arch. Math., 42 (1984), 426–429.MathSciNetCrossRefMATHGoogle Scholar
  64. [64]
    D. E. Dobbs and M. Fontana, Going-up, direct limits and universality, Comm. Math. Univ. St. Pauli, 33 (1984), 191–196.MathSciNetMATHGoogle Scholar
  65. [65]
    D. E. Dobbs and M. Fontana, Kronecker function rings and abstract Riemann surfaces, J. Algebra, 99 (1986), 263–274.MathSciNetCrossRefMATHGoogle Scholar
  66. [66]
    D. E. Dobbs and M. Fontana, Seminormal rings generated by algebraic integers, Mathe-matika, 34 (1987), 141–154.MathSciNetCrossRefMATHGoogle Scholar
  67. [67]
    D. E. Dobbs and M. Fontana, Some results on the weak normalization of an integral domain, Math. J. Okayama U., 31 (1989), 9–23.MathSciNetMATHGoogle Scholar
  68. [68]
    D. E. Dobbs and M. Fontana, Sur les suites dimensionelles et une classe d’anneaux dis­tingués qui les déterminent, C. R. Acad. Sei. Paris A-B, 306 (1988), 11–16.MathSciNetMATHGoogle Scholar
  69. [69]
    D. E. Dobbs and M. Fontana, Integral overrings of two-dimensional going-down domains, Proc. Amer. Math. Soc, 115 (1992), 655–662.MathSciNetCrossRefMATHGoogle Scholar
  70. [70]
    D. E. Dobbs and M. Fontana, Inverse limits of integral domains arising from iterated Nagata composition, Math. Scand., to appear.Google Scholar
  71. [71]
    D. E. Dobbs and M. Fontana, Lifting trees of prime ideals to Bézout extension domains, Comm. Algebra, 27 (1999), 6243–6252.MathSciNetCrossRefMATHGoogle Scholar
  72. [72]
    D. E. Dobbs, M. Fontana, J. A. Huckaba and I. J. Papick, Strong ring extensions and pseudo-valuation domains, Houston J. Math., 8 (1982), 167–184.MathSciNetMATHGoogle Scholar
  73. [73]
    D. E. Dobbs, M. Fontana and S. Kabbaj, Direct limits of Jaffard domains and S-domains, Comm. Math. Univ. St. Pauli, 39 (1990), 143–155.MathSciNetMATHGoogle Scholar
  74. [74]
    D. E. Dobbs, M. Fontana and I. J. Papick, On the flat spectral topology, Rend. Mat., 4 (1981), 559–578.MathSciNetGoogle Scholar
  75. [75]
    D. E. Dobbs, M. Fontana and I. J. Papick, Direct limits and going-down, Comm. Math. Univ. St. Pauli, 31 (1982), 129–135.MathSciNetMATHGoogle Scholar
  76. [76]
    D. E. Dobbs and E. G. Houston, On t-Spec(R[[X]]), Canad. Math. Bull., 38 (1995), 187–195.MathSciNetCrossRefMATHGoogle Scholar
  77. [77]
    D. E. Dobbs, E. G. Houston, T. G. Lucas, M. Roitman and M. Zafrullah, On t-linked overrings, Comm. Algebra, 20 (1992), 1463–1488.MathSciNetCrossRefMATHGoogle Scholar
  78. [78]
    D. E. Dobbs and I. J. Papick, Going-down: a survey, Nieuw Arch. v. Wisk., 26 (1978), 255–291.MathSciNetMATHGoogle Scholar
  79. [79]
    M. Fontana, Topologically defined classes of commutative rings, Ann. Mat. Pura Appl., 123 (1980), 331–355.MathSciNetCrossRefMATHGoogle Scholar
  80. [80]
    M. Fontana, Sur quelques classes d’anneaux divisés, Rend. Sem. Mat. Fis. Milano, 51 (1981), 179–200.MathSciNetCrossRefMATHGoogle Scholar
  81. [81]
    M. S. Gilbert, Extensions of commutative rings with linearly ordered intermediate rings, Ph. D. dissertation, Univ. Tennessee, Knoxville, 1996.Google Scholar
  82. [82]
    R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972.MATHGoogle Scholar
  83. [83]
    F. Girolami, The catenarian property of power series rings over a globalized pseudo-valuation domain, Rend. Circ. Mat. Palermo, 38 (1989), 5–12.MathSciNetCrossRefMATHGoogle Scholar
  84. [84]
    A. Grothendieck and J. A. Dieudonné, Eléments de Géométrie Algébrique, I, Springer-Verlag, Berlin/Heidelberg/New York, 1971.MATHGoogle Scholar
  85. [85]
    J. R. Hedstrom and E. G. Houston, Pseudo-valuation domains, II, Houston J. Math., 4 (1978), 199–207.MathSciNetMATHGoogle Scholar
  86. [86]
    W. Heinzer and S. Wiegand, Prime ideals in two-dimensional polynomial rings, Proc. Amer. Math. Soc, 107 (1989), 577–586.MathSciNetCrossRefMATHGoogle Scholar
  87. [87]
    M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc, 142 (1969), 43–60.MathSciNetCrossRefMATHGoogle Scholar
  88. [88]
    J. A. Huckaba, Commutative Rings with Zero Divisors, Dekker, New York, 1988.MATHGoogle Scholar
  89. [89]
    B. Y. Kang and D. Y. Oh, Lifting up an infinite chain of prime ideals to a valuation ring, Proc. Amer. Math. Soc, 126 (1998), 645–646.MathSciNetCrossRefMATHGoogle Scholar
  90. [90]
    Y. Lequain, Going down and unibranchness, Bol. Soc. Brasil. Mat., 11 (1980), 51–54.MathSciNetCrossRefMATHGoogle Scholar
  91. [91]
    I. Kaplansky, Commutative Rings, rev. ed., Univ. Chicago Press, Chicago, 1974.MATHGoogle Scholar
  92. [92]
    S. Malik and J. L. Mott, Strong S-domains, J. Pure Appl. Algebra, 28 (1983), 249–264.MathSciNetCrossRefMATHGoogle Scholar
  93. [93]
    S. McAdam, Going down in polynomial rings, Canad. Math. J., 23 (1971), 704–711.MathSciNetCrossRefMATHGoogle Scholar
  94. [94]
    M. Nagata, Finitely generated rings over a valuation domain, J. Math. Kyoto Univ., 5 (1966), 163–169.MathSciNetMATHGoogle Scholar
  95. [95]
    A. Okabe and K. Yoshida, Note on strong pseudo-valuation domains, Bull. Fac Sei. Ibaraki Univ. Ser. A , 21 (1989), 9–12.MathSciNetCrossRefMATHGoogle Scholar
  96. [96]
    I. J. Papick, Topologically defined classes of going-down domains, Trans. Amer. Math. Soc, 219 (1976), 1–37.MathSciNetCrossRefMATHGoogle Scholar
  97. [97]
    L. J. Ratliff, Jr., On quasi-unmixed local domains, the altitude formula, and the chain condition for prime ideals, II, Amer. J. Math., 92 (1970), 99–144.MathSciNetCrossRefMATHGoogle Scholar
  98. [98]
    R. G. Swan, On seminormality, J. Algebra, 67 (1980), 210–229.MathSciNetCrossRefMATHGoogle Scholar
  99. [99]
    H. Yanagihara, Some results on weakly normal ring extensions, J. Math. Soc. Japan, 35 (1983), 649–661.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • David E. Dobbs
    • 1
  1. 1.Department of MathematicsThe University of TennesseeKnoxvilleUSA

Personalised recommendations