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Finite Conductor Rings with Zero Divisors

  • Sarah Glaz
Part of the Mathematics and Its Applications book series (MAIA, volume 520)

Abstract

The finite conductor property of a domain R— that is the finite generation of the conductor ideals (I: J) for principal ideals I and J of R, came into prominence with the publication of McAdam’s work [35]. The definition of a finite conductor domain appears in an early unpublished version of McAdam’s manuscript, but it appears in print for the first time in [11]. The notion embodies, in its various aspects, both factoriality properties and finiteness conditions. Indeed, the class of domains where (I: J) is always principal for any two principal ideals I and J, is precisely that of Greatest Common Divisor (GCD) domains, while the requirement that each such (I: J) be finitely generated is a necessary condition for the coherence of a domain. For that reason the finite conductor property makes frequent, explicit or implicit, appearance in the literature in two kinds of, occasionally intermingling, investigations: those involving factoriality and those concerned with finiteness, coherent-like conditions, of domains. Regarding investigations involving factoriality: GCD domains were investigated in their own right for a variety of structural properties, as an aspect of properties of various ring constructions, and as a source of generalizations to other classes of rings. Articles [2, 3, 5, 6, 9, 10, 11, 12, 15, 18, 31, 32, 44, 50] provide just a partial list of references on the subject.

Keywords

Principal Ideal Regular Ring Zero Divisor Minimal Prime Ideal Finite Conductor 
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]
    B. Alfonsi, Grade Non-Noetherien, Comm. in Algebra 9 (1981), 811–840.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    D.D. Anderson and D.F. Anderson, Generalized GCD Domains, Comment. Math.Univ. St. Pauli XXVIII (1979), 215–221.Google Scholar
  3. [3]
    D.D. Anderson and R. Markanda, Unique Factorization Rings With Zero Divisors, Houston J. of Math. 11 (1985), 15–30.MathSciNetzbMATHGoogle Scholar
  4. [4]
    D.D. Anderson, D.F. Anderson and M. Zafrullah, Factorization in Integral Domains, J. of Pure Appl. Algebra 69 (1990), 1–19.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    D.D. Anderson and M. Zafrullah, Almost Bezout Domains, J. of Algebra 142 (1991), 285–309.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    D.D. Anderson, K.R. Knopp and R.L. Lewin, Almost Bezout Domains, II, J. of Algebra 167 (1994), 547–556.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    V. Barucci, D.F. Anderson and D. Dobbs, Coherent Mori Domains and the Principal Ideal Theorem, Comm. in Algebra 15 (1987), 1119–1156.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    G.M. Bergman, Groups acting on hereditary rings, Proc. London Math. Soc. 23 (1971), 70–82.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    N. Bourbaki, “Commutative Algebra”, Addison Wesley, 1972.zbMATHGoogle Scholar
  10. [10]
    P.M. Cohn, Bezout Rings and Their Subrings, Proc. Camb. Phil. Soc. 64 (1968), 251–264.zbMATHCrossRefGoogle Scholar
  11. [11]
    D. Dobbs, On Going Down for Simple Overrings, Proc. AMS 39 (1973), 515–519.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    D. Dobbs and I. Papick, When is D+M Coherent? Proc. AMS 56 (1976), 51–54.MathSciNetzbMATHGoogle Scholar
  13. [13]
    D. Dobbs, On Flat Divided Prime Ideals, in: “Factorization in Integral Domains”, Marcel Dekker Lecture Notes in Math # 189 (1997), 305–315.Google Scholar
  14. [14]
    P. Eakin and W. Heinzer, Non-Finiteness of Finite Dimensional Krull Domains, J. of Algebra 14 (1970), 333–340.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    S. Gabelli and E. Houston, Coherent-Like Conditions in Pullbacks, Mich. Math. J. 44 (1997). 99–123.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    R. Gilmer, “Multiplicative Ideal Theory”, Queens Papers in Pure and Apll. Math. 12, 1968.zbMATHGoogle Scholar
  17. [17]
    R. Gilmer, A Two Dimensional Non-Noetherian Factorial Ring, Proc. AMS 44 (1974), 25–30.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    R. Gilmer and T. Parker, Divisibility Properties in Semigroup Rings, Mich. Math. J. 21 (1974), 65–86.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    R. Gilmer, “Commutative Semigroup Rings”, Chicago Lectures in Math.,1984.zbMATHGoogle Scholar
  20. [20]
    S. Glaz, On The Weak Dimension of Coherent Group Rings, Comm. in Algebra 15 (1987), 1841–1858.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    S. Glaz, Factoriality and Finiteness Properties of Subalgebras over which k[x1,…, xn] is Faithfully Flat, Comm. in Algebra 16 (1988), 1791–1811.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    S. Glaz, “Commutative Coherent Rings”, Springer-Verlag Lecture Notes in Math. 1371, 1989.zbMATHGoogle Scholar
  23. [23]
    S. Glaz, On the Coherence and Weak Dimensions of the Rings R<x> and R(x), Proc. AMS 106 (1989), 579–598.MathSciNetzbMATHGoogle Scholar
  24. [24]
    S. Glaz, Fixed Rings of Coherent Regular Rings, Comm. in Algebra 20 (1992), 2635–2651.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    S. Glaz, Finite Conductor rings, to appear in Proc. Amer. Math. Soc. Google Scholar
  26. [26]
    L. Gruson and M. Raynaud, Criteres de platitude et de projectivite, Inv. Math. 13 (1971), 1–89.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    M. Harris, Some Results on Coherent Rings II, Glasgow Math. J. 8 (1967), 123–126.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    W. Heinzer and D. Lantz, The Laskerian Property in Commutative Rings, J. of Algebra 72 (1981), 101–114.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    W. Heinzer and D. Lantz, Commutative Rings with ACC on n-Generated Ideals, J. of Algebra 80 (1983), 261–278.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    S. Jondrup, Groups Acting on Rings, J. London Math. Soc. 8 (1974), 483–486.MathSciNetCrossRefGoogle Scholar
  31. [31]
    I. Kaplansky, “Commutative Rings”, Allyn and Bacon, 1970.zbMATHGoogle Scholar
  32. [32]
    G. Karpilovsky, “Commutative Group Algebras”, Mono. in Pure and Apll. Math. 78. Marcel Dekker, 1983.zbMATHGoogle Scholar
  33. [33]
    E. Matlis, The Minimal Spectrum of a Reduced Ring, Ill. J. of Math. 27 (1983), 353–391.MathSciNetzbMATHGoogle Scholar
  34. [34]
    H. Matsumura, “Commutative Ring Theory”, Cambridge Stud. in Adv. Math. 8, 1986.zbMATHGoogle Scholar
  35. [35]
    S. Mc Adam, Two Conductor Theorems, J. of Algebra 23 (1972), 239–240.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    J. Mott and M. Zafrullah, On Prufer v-Multiplication Domains, Manuscripta Math. 35 (1981), 1–26.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    K. Nagarajan, Groups Acting on Noetherian Rings, Nieuw Arch. voor Wisk. XVI (1968), 25–29.MathSciNetGoogle Scholar
  38. [38]
    M. Nagata, Some Remarks on Prime Divisors, Mem. Univ. of Kyoto, Ser. A 33 (1960), 297–299.MathSciNetzbMATHGoogle Scholar
  39. [39]
    D. Nour El Abidine, Sur un Theoreme de Nagata, Comm. in Algebra 20 (1992), 2127–2138.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    Y. Quentel, Sur la Compacite du Spectre Minimal d’un Anneau, Bull. Soc. Math. France 99 (1971), 265–272.MathSciNetzbMATHGoogle Scholar
  41. [41]
    Y. Quentel, Erratum, Sur la Compacite du Spectre Minimal d’un Anneau, Bull. Soc. Math. France 100 (1972), 461.MathSciNetGoogle Scholar
  42. [42]
    Y. Quentel, Sur L’uniforme Coherence des Anneaux Noetheriens, C.R.Acad.Sc. Paris, Ser. A 275 (1972), 753–755.MathSciNetzbMATHGoogle Scholar
  43. [43]
    J. Querre, Ideaux Divisoriels d’un Anneau de Polynomes, J. of Algebra 64 (1980), 270–284.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    P.B. Sheldon, Prime Ideals in GCD Domains, Can. Math. J. XXVI (1974), 98–107.MathSciNetCrossRefGoogle Scholar
  45. [45]
    J-P. Soublin. Un Anneau Coherent Don’t L’Anneau de Polynomes N’est Pas Coherent, C. R. Acad. Sc. Paris, Ser. A 269 (1968), 497–499.MathSciNetGoogle Scholar
  46. [46]
    B. Stenstrom, “Rings of Quotients”, Springer Verlag, 1975.CrossRefGoogle Scholar
  47. [47]
    W.V. Vasconcelos, “Divisor Theory in Module Categories”, North Holland Math. Studies 14, 1974.zbMATHGoogle Scholar
  48. [48]
    W. Vasconcelos, “The Rings of Dimension Two”, Lecture Notes in Pure and Appl. Math. 22, Marcel Dekker, 1976.Google Scholar
  49. [49]
    M. Zafrullah, On Finite Conductor Domains, Manuscripta Math. 24 (1978), 191–204.MathSciNetzbMATHCrossRefGoogle Scholar
  50. [50]
    M. Zafrullah, A General Theory of Almost Factoriality, Manuscripta Math. 51 (1985), 29–62.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Sarah Glaz
    • 1
  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA

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