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The Class Group and Local Class Group of an Integral Domain

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

Abstract

This article gives a survey of recent developments on the (t-)class group and local (t-)class group of an integral domain R. Let T (R) be the group of t-invertible (fractional) t-ideals of R under t-multiplication, and let Prin (R) (resp., Inv (R)) be its subgroup of principal (resp., invertible) (fractional) ideals. Then Cl (R) = T (R)/Prin (R) is an abelian group, called the (t-)class group of R; the Picard group of R is Pic (R) = Inv (R)/Prin (R); and the local (t-)class group of R is G (R) = T (R)/Inv (R) = Cl (R)/Pic (R). If R is a Krull domain, then Cl (R) is the usual divisor class group of R, and if R is a Prüfer domain, then Cl (R) (= Pic (R)) is just the ideal class group of R.

Keywords

Class Group Integral Domain Valuation Domain Mori Domain Ideal Class Group 
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.

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References

  1. [1]
    D. D. Anderson, 7r-domains, overrings, and divisorial ideals, Glasgow Math. J. 19 (1978), 199–203.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    D. D. Anderson, Globalization of some local properties in Krull domains, Proc. Amer. Math. Soc. 85 (1982), 141–145.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    D. D. Anderson and D. F. Anderson, Generalized GCD domains, Comment. Math. Univ. St. Paul. 28 (1979), 215–221.Google Scholar
  4. [4]
    D.D. Anderson and D. F. Anderson, Divisorial ideals and invertible ideals in graded integral domains, J. Algebra 76 (1982), 549–569.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    D. D. Anderson and D. F. Anderson, Some remarks on star operations and the class group, J. Pure Appl. Algebra 51 (1988), 27–33.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    D. D. Anderson and D. F. Anderson, Elasticity of factorizations in integral domains, J. Pure Appl. Algebra 80 (1992), 217–235.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    D. D. Anderson and D. F. Anderson, The ring R[X, r/X], Lecture Notes in Pure and Applied Mathematics, vol. 171, Marcel Dekker, New York, 1995, 95–113.Google Scholar
  8. [8]
    D. D. Anderson, D. F. Anderson, and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra 69 (1990), 1–19.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    D. D. Anderson, D. F. Anderson, and M. Zafrullah, Rings between D[X] and D[X], Houston J. Math. 17 (1991), 109–129.MathSciNetMATHGoogle Scholar
  10. [10]
    D. D. Anderson, D. F. Anderson, and M. Zafrullah, Splitting the t-class group, J. Pure Appl. Algebra 74 (1991), 17–37.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    D. D. Anderson, D. F. Anderson, and M. Zafrullah, Factorization in integral domains, II, J. Algebra 152 (1992), 78–93.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    D. D. Anderson, E. G. Houston, and M. Zafrullah, T-linked extensions, the t-class group, and Nagata’s theorem, J. Pure Appl. Algebra 86 (1993), 109–124.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    D. D. Anderson and L. A. Mahaney, On primary factorizations, J. Pure Appl. Algebra 54 (1988), 141–154.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    D. D. Anderson, J. L. Mott, and M. Zafrullah, Finite character representations for integral domains, Boll. Un. Math. Ital. (7) 6-B (1992), 613–630.MathSciNetGoogle Scholar
  15. [15]
    D. D. Anderson and M. Zafrullah, Weakly factorial domains and groups of divisibility, Proc. Amer. Math. Soc. 109 (1990), 907–913.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    D. D. Anderson and M. Zafrullah, P. M. Cohn’s completely primal elements, Lecture Notes in Pure and Applied Mathematics, vol. 171, Marcel Dekker, New York, 1995, 115–123.MathSciNetGoogle Scholar
  17. [17]
    D. F. Anderson, Subrings of K[X, Y] generated by monomials, Canad. J. Math. 20 (1978), 215–224.Google Scholar
  18. [18]
    D. F. Anderson, Graded Krull domains, Comm. Algebra 7 (1979), 79–106.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    D. F. Anderson, A general theory of class groups, Comm. Algebra 16 (1988), 805–847.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    D. F. Anderson, Elasticity of factorizations in integral domains, a survey, Lecture Notes in Pure and Applied Mathematics, vol. 189, Marcel Dekker, New York, 1997, 1–29.Google Scholar
  21. [21]
    D. F. Anderson, S. E. Baghdadi, and S.-E. Kabbaj, On the class group of A + XB[X] domains, Lecture Notes in Pure and Applied Mathematics, vol. 205, Marcel Dekker, New York, 1999, 73–85.Google Scholar
  22. [22]
    D. F. Anderson, S. E. Baghdadi, and S.-E. Kabbaj, The homogeneous class group of A + Xß[X] domains, preprint.Google Scholar
  23. [23]
    D. F. Anderson and A. Bouvier, Ideal transforms and overrings of a quasilocal integral domain, Ann. Univ. Ferrara-Sez. VII, 32 (1986), 15–38.MathSciNetGoogle Scholar
  24. [24]
    D. F. Anderson and D. Nour El Abidine, Some remarks on the ring R#, Lecture Notes in Pure and Applied Mathematics, vol. 185, Marcel Dekker, New York, 1997, 33–43.Google Scholar
  25. [25]
    D. F. Anderson and D. Nour El Abidine, Factorization in integral domains, III, J. Pure Appl. Algebra 135 (1999), 107–127.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    D. F. Anderson and D. Nour El Abidine, The A + XB[X] and A + *B[[A”]] constructions from GCD-domains, J. Pure Appl. Algebra, to appear.Google Scholar
  27. [27]
    D. F. Anderson and A. Ryckaert, The class group of D + M, J. Pure Appl. Algebra 52 (1988), 199–212.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    J. T. Arnold and J. W. Brewer, On flat overrings, ideal transforms and generalized transforms of a commutative ring, J. Algebra 108 (1987), 161–173.MathSciNetCrossRefGoogle Scholar
  29. [29]
    V. Barucci, D. F. Anderson, and D. E. Dobbs, Coherent Mori domains and the principal ideal theorem, Comm. Algebra 15 (1989), 1119–1156.MathSciNetCrossRefGoogle Scholar
  30. [30]
    V. Barucci and S. Gabelli, On the class group of a Mori domain, J. Algebra 108 (1987), 161–173.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    V. Barucci and S. Gabelli, How far is a Mori domain from being a Krull domain?, J. Pure Appl. Algebra 45 (1987), 101–112.MathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    V. Barucci, S. Gabelli, and M. Roitman, On semi-Krull domains, J. Algebra 145 (1992), 306–328.MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    V. Barucci, S. Gabelli, and M. Roitman, The class group of a strongly Mori domain, Comm. Algebra 22 (1994), 173–211.MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    V. Barucci, L. Izelgue, and S.-E. Kabbaj, Some factorization properties of A + XB[X] domains, Lecture Notes in Pure and Applied Mathematics, vol. 185, Marcel Dekker, New York, 1997, 69–78.MathSciNetGoogle Scholar
  35. [35]
    E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form D + M, Michigan Math. J. 20 (1973), 79–95.MathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    A. Bouvier, Survey of locally factorial Krull domains, Pub. Dept. Math. Lyon 17–1 (1980), 1–31.MathSciNetGoogle Scholar
  37. [37]
    A. Bouvier, Le groupe des classes d’un anneau intégré, 107 ème Congrès National des Société Savantes, Brest, Fasc. IV (1982), 85–92.Google Scholar
  38. [38]
    A. Bouvier, The local class group of a Krull domain, Canad. Math. Bull. 26 (1983), 13–19.MathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    A. Bouvier and M. Zafrullah, On some class groups of an integral domain, Bull. Soc. Math. Grèce 29 (1988), 45–59.MathSciNetMATHGoogle Scholar
  40. [40]
    J. Brewer and E. A. Rutter, D + M constructions with general overrings, Michigan Math. J. 23 (1976), 33–42.MathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    P.-J. Cahen and J.-L. Chabert, Integer-valued Polynomials, Amer. Math. Soc. Surveys and Monographs 48, Providence, 1997.MATHGoogle Scholar
  42. [42]
    P.-J. Cahen, S. Gabelli, and E. Houston, Mori domains of integer-valued polynomials, J. Pure Appl. Algebra, to appear.Google Scholar
  43. [43]
    L. Claborn, Every abelian group is a class group, Pacific J. Math. 18 (1966), 219–222.MathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    P. M. Cohn, Bézout rings and their subrings, Math. Proc. Cambridge Philos. Soc. 64 (1968), 251–264.CrossRefMATHGoogle Scholar
  45. [45]
    D. Costa, J. L. Mott, and M. Zafrullah, The construction D + XD S [X], J. Algebra 53 (1978), 423–439.MathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    D. E. Dobbs, E. G. Houston, T. G. Lucas, and M. Zafrullah, t-linked overrings and Prüfer -multiplication domains, Comm. Algebra 17 (1989), 2835–2852.MathSciNetCrossRefMATHGoogle Scholar
  47. [47]
    D. E. Dobbs, E. G. Houston, T. G. Lucas, M. Roitman, and M. Zafrullah, On t-linked overrings, Comm. Algebra 29 (1992), 1463–1488.MathSciNetCrossRefGoogle Scholar
  48. [48]
    T. Dumitrescu, S. O. Ibrahim Al-Salihi, N. Radu, and T. Shah, Some factorization properties of composite domains A + XB[X] and A + XB[[X]], Comm. Algebra, to appear.Google Scholar
  49. [49]
    M. Fontana, Topologically defined classes of commutative rings, Ann. Math. Pura Appl. 123 (1980), 331–355.MathSciNetCrossRefMATHGoogle Scholar
  50. [50]
    M. Fontana and S. Gabelli, On the class group and local class group of a pullback, J. Algebra 181 (1996), 803–835.MathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    M. Fontana and S. Gabelli, Prüfer domains with class group generated by the classes of invertible maximal ideals, Comm. Algebra 25 (1997), 3993–4008.MathSciNetCrossRefMATHGoogle Scholar
  52. [52]
    M. Font ana, J. A. Huckaba, and I. J. Papick, Prüfer Domains, Marcel Dekker, New York, 1997.MATHGoogle Scholar
  53. [53]
    R. Fossum, The Divisor Class Group of a Krull Domain, Springer-Verlag, New York, 1973.CrossRefMATHGoogle Scholar
  54. [54]
    S. Gabelli, On divisorial ideals in polynomial rings over Mori domains, Comm. Algebra 15 (1987), 2349–2370.MathSciNetCrossRefMATHGoogle Scholar
  55. [55]
    S. Gabelli, Divisorial ideals and class groups of Mori domains, Lecture Notes in Pure and Applied Mathematics, vol. 153, Marcel Dekker, New York,1994, 131–139.MathSciNetGoogle Scholar
  56. [56]
    S. Gabelli, On Nagata’s theorem for the class group, II, Lecture Notes in Pure and Applied Mathematics, vol. 206, Marcel Dekker, New York, 1999, 117–142.MathSciNetGoogle Scholar
  57. [57]
    S. Gabelli and M. Roitman, On Nagata’s Theorem for the class group, J. Pure Appl. Algebra 66 (1990), 31–42.MathSciNetCrossRefMATHGoogle Scholar
  58. [58]
    S. Gabelli and F. Tartarone, On the class group of integer-valued polynomial rings over Krull domains, J. Pure Appl. Algebra, to appear.Google Scholar
  59. [59]
    R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers in Pure and Applied Mathematics 12, Queen’s University, Kingston, 1968.MATHGoogle Scholar
  60. [60]
    R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972.MATHGoogle Scholar
  61. [61]
    R. Gilmer and R. Heitmann, On Pic (R[X]) for R seminormal, J. Pure Appl. Algebra 16 (1980), 251–257.MathSciNetCrossRefMATHGoogle Scholar
  62. [62]
    R. Gilmer and T. Parker, Divisibility properties in semigroup rings, Michigan Math. J. 21 (1974), 65–86.MathSciNetCrossRefMATHGoogle Scholar
  63. [63]
    M. Griffin, Some results on A-multiplication rings, Canad. J. Math. 19 (1967), 710–722.MathSciNetCrossRefMATHGoogle Scholar
  64. [64]
    F. Halter-Koch, Elasticity of factorizations in atomic monoids and integral domains, J. Théorie des Nombres Bordeaux 7 (1995), 367–385.MathSciNetCrossRefMATHGoogle Scholar
  65. [65]
    F. Halter-Koch, Divisor theories with primary elements and weakly Krull domains, Boll. Un. Math. Ital. (7) 9-B (1995), 417–441.MathSciNetGoogle Scholar
  66. [66]
    F. Halter-Koch, Finitely generated monoids, finitely primary monoids, and factorization properties of integral domains, Lecture Notes in Pure and Applied Mathematics, vol. 189, Marcel Dekker, New York, 1997, 31–72.MathSciNetGoogle Scholar
  67. [67]
    F. Halter-Koch, Ideal Systems, An Introduction to Multiplicative Ideal Theory, Marcel Dekker, New York, 1998.MATHGoogle Scholar
  68. [68]
    J. Hedstrom and E. Houston, Some remarks on star-operations, J. Pure Appl. Algebra 18 (1980), 37–44.MathSciNetCrossRefMATHGoogle Scholar
  69. [69]
    E. Houston, S. Malik, and J. Mott, Characterizations of A-multiplication domains, Canad. Math. Bull. 27 (1984), 48–52.MathSciNetCrossRefMATHGoogle Scholar
  70. [70]
    P. Jaffard, Les Systèmes d’Idéaux, Dunod, Paris, 1960.MATHGoogle Scholar
  71. [71]
    M. Khalis and D. Nour El Abidine, On the class group of a pullback, Lecture Notes in Pure and Applied Mathematics, vol. 185, Marcel Dekker, New York, 1997, 377–386.Google Scholar
  72. [72]
    F. Kainrath, A note on quotients formed by unit groups of semilocal rings, Houston J. Math. 1998 (124), 613–618.MathSciNetGoogle Scholar
  73. [73]
    F. Kainrath, A divisor theoretic approach towards the arithmetic of noetherian domains, Arch. Math., to appear.Google Scholar
  74. [74]
    F. Kainrath, Elasticity of finitely generated domains, preprint.Google Scholar
  75. [75]
    B. G. Kang, Prüfer v-multiplication domains and the ring R[X]n v, J. Algebra 124 (1989), 151–170.CrossRefGoogle Scholar
  76. [76]
    B. G. Kang, Divisibility properties of group rings over torsion-free abelian groups, Comment. Math. Univ. St. Paul. 48 (1999), 19–24.MathSciNetMATHGoogle Scholar
  77. [77]
    I. Kaplansky, Commutative Rings, Univ. of Chicago Press, Chicago, rev. ed., 1974.MATHGoogle Scholar
  78. [78]
    H. Kim and Y. S. Park, Divisibility properties of semigroup rings over torsion-free cancella-tive semigroups, preprint.Google Scholar
  79. [79]
    S. Malik, J. L. Mott, and M. Zafrullah, On t-invertibility, Comm. Algebra 16 (1988), 149–170.MathSciNetCrossRefMATHGoogle Scholar
  80. [80]
    J. L. Mott and M. Schexnayder, Exact sequences of semivalue groups, J. Reine Angew. Math. 283/284 (1976), 388–401.MathSciNetGoogle Scholar
  81. [81]
    J. L. Mott and M. Zafrullah, On Prüfer A-multiplication domains, Manuscripta Math. 35 (1981), 1–26.MathSciNetCrossRefMATHGoogle Scholar
  82. [82]
    M. Nagata, A remark on the unique factorization theorem, J. Math. Soc. Japan 9 (1957), 143–145.MathSciNetCrossRefMATHGoogle Scholar
  83. [83]
    T. Nishimura, On the v-ideal of an integral domain, Bull. Kyoto Gakugei Univ. ser. B 17 (1961), 47–50.Google Scholar
  84. [84]
    D. Nour El Abidine, Groupe des classes d’un anneau intégre, Ann. Univ. Ferrara-Sez VII, 36 (1990), 175–183.Google Scholar
  85. [85]
    D. Nour El Abidine, Sur le groupe des classes d’un anneau intégre, C. R. Math. Rep. Acad. Sei. Canada 13 (1991), 69–74.Google Scholar
  86. [86]
    D. Nour El Abidine, Groupe des Classes de Certains Anneaux Intéges et Idéaux Transformés, Thèse de doctorat de l’Université Claude Bernard-Lyon I (1992).Google Scholar
  87. [87]
    D. Nour El Abidine, Sur un théorème de Nagata, Comm. Algebra 20 (1992), 2127–2138.MathSciNetCrossRefMATHGoogle Scholar
  88. [88]
    D. Nour El Abidine, Groupe des classes de l’algèbre affine d’un hyperplan, Bull. Soc. Math. Grèce 34 (1992), 1–12.Google Scholar
  89. [89]
    D. Nour El Abidine, Groupe des classes anneaux de type A[X, Y]/ (aX + bY), Boll. Un. Mat. Ital. (7), 6-A (1992), 271–280.Google Scholar
  90. [90]
    D. Nour El Abidine, Groupe des classes des anneaux de polynômes sur un D + M, C. R. Math. Rep. Acad. Sei. Canada 18 (1996), 189–192.MATHGoogle Scholar
  91. [91]
    A. Ryckaert, Sur le Groupe des Classes et le Groupe Locale des Classes d’un Anneau Integre, These de doctorat de l’Université Claude Bernard-Lyon I (1986).Google Scholar
  92. [92]
    A. Ryckaert, Sur le groupe des classes des anneaux “D+M”, C. R. Math. Rep. Acad. Sei. Canada 9 (1986), 363–367.MathSciNetGoogle Scholar
  93. [93]
    P. Samuel, Lectures on Unique Factorization Domains, Tata Inst, for Fundamental Research, Bombay, No. 30, 1964.Google Scholar
  94. [94]
    P. Samuel, Unique factorization, Amer. Math. Monthly 75 (1968), 945–952.MathSciNetCrossRefMATHGoogle Scholar
  95. [95]
    U. Storch, Fastfaktorielle Ringe, Schriftenreihe Math. Inst. Univ. Münster, Heft 36 (1967).MATHGoogle Scholar
  96. [96]
    R. J. Valenza, Elasticity of factorizations in number fields, J. Number Theory 36 (1990), 212–218.MathSciNetCrossRefMATHGoogle Scholar
  97. [97]
    M. Zafrullah, On finite conductor domains, Manuscripta Math. 24 (1978), 191–204.MathSciNetCrossRefMATHGoogle Scholar
  98. [98]
    M. Zafrullah, A general theory of almost factoriality, Manuscripta Math. 51 (1985), 29–62.MathSciNetCrossRefMATHGoogle Scholar
  99. [99]
    M. Zafrullah, On a property of pre-Schreier domains, Comm. Algebra 15 (1987), 1895–1920.MathSciNetCrossRefMATHGoogle Scholar
  100. [100]
    M. Zafrullah, Putting t-invertibility to use, this volume.Google Scholar
  101. [101]
    A. Zaks, Half-factorial domains, Israel J. Math. 37 (1980), 281–302.MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • David F. Anderson
    • 1
  1. 1.Mathematics DepartmentThe University of TennesseeKnoxvilleUSA

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