Advertisement

Half-Factorial Domains, a Survey

  • Scott T. Chapman
  • Jim Coykendall
Part of the Mathematics and Its Applications book series (MAIA, volume 520)

Abstract

Let D be an integral domain. D is atomic if every nonzero nonunit of D can be written as a product of irreducible elements (or atoms) of D. Let 1 (D) represent the set of irreducible elements of D. Traditionally, an atomic domain D is a unique factorization domain (UFD) if α 1α n = β 1β m for each ai and β jI (D) implies:
  1. 1.

    n =m,

     
  2. 2.

    there exists a permutation б of {1,... ,n} such that α 1 and β б (i) are associates.

     

Keywords

Prime Ideal Class Group Integral Domain Class Number Integral Closure 
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, Factorization in Integral Domains, Lecture Notes in Pure and Applied Math­ematics, 189 (1997), Marcel Dekker, New York.zbMATHGoogle Scholar
  2. [2]
    D.D. Anderson, D.F. Anderson and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra 69 (1990), 1–19.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    D.D. Anderson, D.F. Anderson and M. Zafrullah, Rings between D[X] and if [X], Houston J. Math. 17 (1991), 109–129.MathSciNetzbMATHGoogle Scholar
  4. [4]
    D.D. Anderson, D.F. Anderson and M. Zafrullah, Factorization in integral domains, II, J. Algebra 152 (1992), 78–93.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    D.D. Anderson and J.L. Mott, Cohen-Kaplansky domains: integral domains with a finite number of irreducible elements, J. Algebra 148 (1992), 17–41.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    D.F. Anderson, Elasticity of factorizations in integral domains: a survey, Factorization in Integral Domains, Lecture Notes in Pure and Applied Mathematics, 189 (1997), Marcel Dekker, New York, 1–30.Google Scholar
  7. [7]
    D.F. Anderson, S.T. Chapman and W.W. Smith, Some factorization properties of Krull domains with infinite cyclic divisor class group, J. Pure Appl. Algebra 96 (1994), 97–112.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    D.F. Anderson, S.T. Chapman and W.W. Smith, On Krull half-factorial domains with infi­nite cyclic divisor class group, Houston J. Math. 20 (1994), 561–570.MathSciNetzbMATHGoogle Scholar
  9. [9]
    D.F. Anderson, S.T. Chapman and W.W. Smith, Overrings of half-factorial domains, Canad. Math. Bui. 37 (1994), 437–442.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    D.F. Anderson, S.T. Chapman and W.W. Smith, Overrings of half-factorial domains, II, Comm. Algebra 23 (1995), 3961–3976.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    D.F. Anderson and D. Nour El Abidine, Factorization in integral domains, III, J. Pure Appl. Algebra 135 (1999), 107–127.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    D.F. Anderson and J. Park, Locally half-factorial domains, Houston J. Math. 23 (1997), 617–630.MathSciNetzbMATHGoogle Scholar
  13. [13]
    D.F. Anderson and J. Park, Factorization in subrings of K[X] or Ä”[[X]], Factorization in Integral Domains, Lecture Notes in Pure and Applied Mathematics, 189 (1997), Marcel Dekker, New York, 227–242.MathSciNetGoogle Scholar
  14. [14]
    D.F. Anderson and J. Winner, Factorization in X[[5]], Factorization in Integral Domains, Lecture Notes in Pure and Applied Mathematics, 189 (1997), Marcel Dekker, New York, 243–256.MathSciNetGoogle Scholar
  15. [15]
    V. Barucci, L. Izelgue and S.E. Kabbaj, Some factorization properties of A-\-XB[X] domains, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 185 (1997), 69–78.MathSciNetGoogle Scholar
  16. [16]
    L. Carlitz, A characterization of algebraic number fields with class number two. Proc. Amer. Math. Soc. 11 (1960), 391–2.MathSciNetzbMATHGoogle Scholar
  17. [17]
    S. T. Chapman, On the Davenport constant, the cross number and their application in factorization theory, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 189 (1995), 167–190.Google Scholar
  18. [18]
    S. T. Chapman, M. Freeze and W. W. Smith, On generalized lengths of factorizations in Dedekind and Krull domains, this volume.Google Scholar
  19. [19]
    S.T. Chapman and A. Geroldinger, Krull domains and monoids, their sets of lengths and associated combinatorial problems, Lecture notes in Pure and Applied Mathematics, Marcel-Dekker, 189 (1997), 73–112.MathSciNetGoogle Scholar
  20. [20]
    S.T. Chapman and W.W. Smith, Factorization in Dedekind domains with finite class group, Israel J. Math 71 (1990), 65–95.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    S.T. Chapman and W.W. Smith, On the HFD, CHFD and k-HFD properties in Dedekind domains, Comm. Algebra 20 (1992), 1955–1987.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    S.T. Chapman and W.W. Smith, An analysis using the Zaks-Skula constant of element factorizations in Dedekind domains, J. Algebra 159 (1993), 176–190.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    J. Coykendall, A characterization of polynomial rings with the half-factorial property, Lec­ture Notes in Pure and Applied Mathematics, Marcel Dekker, 189 (1997), 291–294.MathSciNetGoogle Scholar
  24. [24]
    J. Coykendall, The half-factorial property in integral extensions,Comm. Algebra 27 (1999), 3153–3159.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    J. Coykendall, Half-factorial domains in quadratic fields, submitted.Google Scholar
  26. [26]
    J. Coykendall, A counterexample relating to the integral closure of a half-factorial domain, submitted.Google Scholar
  27. [27]
    P. Erdös and A. Zaks, Reducible sums and splittable sets, J. Number Theory 36 (1990), 89–94.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    W. D. Gao and A. Geroldinger, Half-factorial domains and half-factorial subsets in finite abelian groups, Houston J. Math. 24 (1998), 593–611.MathSciNetzbMATHGoogle Scholar
  29. [29]
    A. Geroldinger, Über nicht-eindeutige Zerlegunen in irreduzible Elements, Math. Z. 197 (1988), 505–529.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    A. Geroldinger, The Cross Number of finite abelian groups. Journal of Number Theory 48 (1994), 219–223.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    A. Geroldinger and R. Schneider, On Davenport’s Constant, J. Combin. Theory Ser. A 61 (1992), 147–152.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    A. Geroldinger and R. Schneider, The cross number of finite abelian groups II, Europ. J. Combinatorics 15 (1994), 399–405.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    A. Geroldinger and R. Schneider, The cross number of finite abelian groups III, Discrete Math. 150 (1996), 123–130.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers Pure Appl. Math. Vol. 90, Kingston, Ontario, 1992.zbMATHGoogle Scholar
  35. [35]
    R. Gilmer, W. Heinzer and W.W. Smith, On the distribution of prime ideals within the ideal class group, Houston J. Math. 22 (1996), 51–59.MathSciNetzbMATHGoogle Scholar
  36. [36]
    N. Gonzalez, These de doctorat de l’Université de Droit, d’Economie et des Sciences d’Aix-Marseille (1997).Google Scholar
  37. [37]
    N. Gonzalez, Elasticity and ramification, Comm. Algebra 27 (1999), 1729–1736.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    N. Gonzalez, Elasticity of A + XB[X] domains, J. Pure Appl. Algebra 138 (1999), 119–137.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    A. Grams, The distribution of prime ideals of a Dedekind domain, Bull. Austral. Math. Soc. 11 (1974), 429–441.MathSciNetCrossRefGoogle Scholar
  40. [40]
    F. Halter-Koch, Factorization of Algebraic Integers, Ber. Math. Stat. Sektion im Forschungszentrum 191 (1983).Google Scholar
  41. [41]
    F. Halter-Koch, Halbgruppen mit Divisorentheorie, Expo. Math. 8 (1990), 27–66.MathSciNetzbMATHGoogle Scholar
  42. [42]
    F. Halter-Koch, Finitely generated monoids, finitely primary moniods, and factorization properties of integral domains, Factorization in Integral Domains, Lecture Notes in Pure and Applied Mathematics, 189 (1997), Marcel Dekker, New York, 31–72.MathSciNetGoogle Scholar
  43. [43]
    F. Kainrath, Factorization in Krull monoids with infinite class group, Colloq. Math. 80 (1999), 23–30.MathSciNetzbMATHGoogle Scholar
  44. [44]
    H. Kim, Examples of half-factorial domains, preprint.Google Scholar
  45. [45]
    U. Krause and C. Zahlten, Arithmetic in Krull monoids and the cross number of divisor class groups, Mitteilungen der Mathematischen Gesellschaß in Hamburg 12 (1991), 681–96.MathSciNetzbMATHGoogle Scholar
  46. [46]
    D. Michel and J. Steffan, Repartition des idéaux premiers parmi les classes d’idéaux dans un anneau de Dedekind et équidécomposition, J. Algebra 98 (1986), 82–94.MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    W. Narkiewicz, Some unsolved problems, Bull. Soc. Math. France 25 (1971), 159–164.MathSciNetzbMATHGoogle Scholar
  48. [48]
    W. Narkiewicz, Elementary and analytic theory of algebraic numbers, Springer-Verlag, 1990.zbMATHGoogle Scholar
  49. [49]
    L. Skula, On c-semigroups, Acta Arith. 31 (1976), 247–257.MathSciNetzbMATHGoogle Scholar
  50. [50]
    W. W. Smith, A covering condition for prime ideals, Proc. Amer. Math. Soc. 30 (1971), 451–452.MathSciNetCrossRefGoogle Scholar
  51. [51]
    A. Zaks, Half-factorial domains, Bulletin of the American Mathematical Society 82 (1976), 721–3.MathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    A. Zaks, Half-factorial domains, Israel Journal of Mathematics 37 (1980), 281–302.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Scott T. Chapman
    • 1
  • Jim Coykendall
    • 2
  1. 1.Department of MathematicsTrinity UniversitySan AntonioUSA
  2. 2.Department of MathematicsNorth Dakota State UniversityFargoUSA

Personalised recommendations