Advertisement

Connecting Trace Properties

  • James A. Huckaba
  • Ira Papick
Part of the Mathematics and Its Applications book series (MAIA, volume 520)

Abstract

The study of commutative integral domains often involves the interplay between special collections of ideals and overrings. Structural properties are identified, analyzed, and classified by using the techniques and results of ideal theory. Gilmer’s 1972 book, Multiplicative Ideal Theory, serves as an excellent introduction and foundation to this perspective [G], and the recent book of Fontana, Huckaba, and Papick, Prüfer Domains, continues in this spirit [FHP].

Keywords

Prime Ideal Maximal Ideal Trace Ideal Minimal Prime Ideal Dedekind 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. [AHP]
    D.D. Anderson, J. Huckaba, I. Papick, A note on stable domains, Houston J. Math. 13 (1987), 13–17.MathSciNetzbMATHGoogle Scholar
  2. [B]
    V. Barucci, Strongly divisorial ideals and complete integral closure of an integral domain, J. Algebra 99 (1986), 132–142.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [Ba]
    H. Bass, On the ubiquity of Gorenstein rings, Math Z. 82 (1963), 8–28.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [BG]
    E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form D + M, Michigan Math. J. 20 (1973), 79–95.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [CL]
    P.-J. Cahen and T. Lucas, The special trace property, Commutative Ring Theory: Proceedings of the II International Conference, Marcel Dekker, New York, 1997, 161–172.Google Scholar
  6. [FHP]
    M. Fontana, J. Huckaba, I. Papick, Prüfer Domains, Marcel Dekker, New York, 1997.zbMATHGoogle Scholar
  7. [FHP1]
    M. Fontana, J. Huckaba, I. Papick, Domains satisfying the trace property, J. Algebra 107 (1987), 169–182.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [Fo]
    R. Fossum, The Divisor Class Group of a Krull Domain, Springer-Verlag, New York, 1973.zbMATHCrossRefGoogle Scholar
  9. [Ga]
    S. Gabelli, Domains with the radical trace property and their complete integral closure, Comm. Algebra 20 (1992), 829–845.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [G]
    R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972.zbMATHGoogle Scholar
  11. [GHe]
    R. Gilmer and W. Heinzer, Overrings of Prüfer rings II, J. Algebra 7 (1967), 281–302.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [GHu]
    R. Gilmer and J. Huckaba, The transform formula for ideals, J. Algebra 21 (1972), 191–215MathSciNetzbMATHCrossRefGoogle Scholar
  13. [HH]
    J. Hedstrom and E. Houston, Pseudo-valuation domains, Pacific. J. Math. 75 (1978), 137–147.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [HeP]
    W. Heinzer and I. Papick, The radical trace property, J. Algebra 112 (1988), 110–121.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [HuP]
    J. Huckaba and I. Papick, When the dual of an ideal is a ring, Manuscripta Math. 37 (1982), 67–85.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [KLM 1]
    S. Kabbaj, T. Lucas, and A. Mimouni, Trace properties and integral domains, Advances in Commutative Ring Theory, 205 (1999), 421–436.MathSciNetGoogle Scholar
  17. [KLM 2]
    S. Kabbaj, T. Lucas and A. Mimouni, Trace properties and pullbacks, I, preprint.Google Scholar
  18. [L]
    T. Lucas, The radical trace property and primary ideals, J. Algebra 184 (1996), 1093–1112.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [LM]
    T. Lucas and A. Mimouni, Trace properties and pullbacks, II, preprint.Google Scholar
  20. [O]
    B. Olberding, Globalizing local properties of Prüfer domains, J. Algebra 205 (1998), 480–504.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • James A. Huckaba
    • 1
  • Ira Papick
    • 1
  1. 1.Department of MathematicsUniversity of Missouri-ColumbiaColumbiaUSA

Personalised recommendations