Ten Problems on Stability of Domains

Chapter

Abstract

We survey the notions of (finite) stability, quasi-stability, and Clifford regularity of domains and illustrate some open problems.

Keywords

Stable ideal Divisorial ideal Flat ideal Clifford regular domain 

Mathematics Subject Classification (2011)

13A15 13F05 13C10 

References

  1. 1.
    D.D. Anderson, J.A. Huckaba, I.J. Papick, A note on stable domains. Houston J. Math. 13, 13–17 (1987)MATHMathSciNetGoogle Scholar
  2. 2.
    D.D. Anderson, M. Zafrullah, Integral domains in which nonzero locally principal ideals are invertible. Comm. Algebra 39, 933–941 (2011)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    C. Arf, Une interprétation algébrique de la suite des ordres de mulptiplicité d’une branche algébrique. Proc. London Math. Soc. 50, 256–287 (1949)MathSciNetGoogle Scholar
  4. 4.
    V. Barucci, Mori domains, Non-Noetherian Commutative Ring Theory; Recent Advances, Chapter 3 (Kluwer Academic Publishers, Springer, 2000)Google Scholar
  5. 5.
    H. Bass, On the ubiquity of Gorenstein rings. Math Z. 82, 8–28 (1963)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    S. Bazzoni, Class semigroups of Prüfer domains. J. Algebra 184, 613–631 (1996)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    S. Bazzoni, Divisorial Domains. Forum Math. 12, 397–419 (2000)MATHMathSciNetGoogle Scholar
  8. 8.
    S. Bazzoni, Groups in the class semigroups of Prüfer domains of finite character. Comm. Algebra 28, 5157–5167 (2000)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    S. Bazzoni, Clifford regular domains. J. Algebra 238, 703–722 (2001)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    S. Bazzoni, Finite character of finitely stable domains. J. Pure Appl. Algebra 215, 1127–1132 (2011)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    S. Bazzoni, L. Salce, Groups in the class semigroups of valuation domains. Israel J. Math. 95, 135–155 (1996)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    S. Bazzoni, L. Salce, Warfield domains. J. Algebra 185, 836–868 (1996)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    A. Bouvier, The local class group of a Krull domain. Canad. Math. Bull. 26, 13–19 (1983)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    M. Fontana, J.A. Huckaba, I.J. Papick, Prüfer domains, Monographs and Textbooks in Pure and Applied Mathematics, vol. 203 (M. Dekker, New York, 1997)Google Scholar
  15. 15.
    P. Eakin, A. Sathaye, Prestable ideals. J. Algebra 41, 439–454 (1976)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    C.A. Finocchiaro, G. Picozza, F. Tartarone, Star-Invertibility and t-finite character in Integral Domains. J. Algebra Appl. 10, 755–769 (2011)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    S. Gabelli, A Class of Prüfer Domains with Nice Divisorial Ideals, Commutative Ring Theory (Fés, 1995), Lecture Notes in Pure and Applied Mathematics, 185 (Dekker, New York, 1997) pp. 313–318Google Scholar
  18. 18.
    S. Gabelli, Generalized Dedekind Domains, Multiplicative Ideal Theory in Commutative Algebra. A tribute to Robert Gilmer (Springer, New York, 2006)Google Scholar
  19. 19.
    S. Gabelli, Locally principal ideals and finite character. Bull. Math. Soc. Sci. Math. Roumanie (N.S.), Tome 56(104), 99–108 (2013)Google Scholar
  20. 20.
    S. Gabelli, E. Houston, G. Picozza, w-Divisoriality in polynomial rings. Comm. Algebra 37, 1117–1127 (2009)Google Scholar
  21. 21.
    S. Gabelli, G. Picozza, Star-stable domains. J. Pure Appl. Algebra 208, 853–866 (2007)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    S. Gabelli, G. Picozza, Star stability and star regularity for Mori domains. Rend. Semin. Mat. Padova 126, 107–125 (2011)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    S. Gabelli, G. Picozza, Stability and regularity with respect to star operations. Comm. Algebra 40, 3558–3582 (2012)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    S. Gabelli, M. Roitman, On finitely Stable Domains (submitted)Google Scholar
  25. 25.
    W. Heinzer, Integral domains in which each non-zero ideal is divisorial. Matematika 15, 164–170 (1968)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    F. Halter-Koch, Clifford semigroups of ideals in monoids and domains. Forum Math. 21, 1001–1020 (2009)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    W. Heinzer, D. Lantz, K. Shah, The Ratliff-Rush ideals in a Noetherian ring. Comm. Algebra 20, 591–622 (1992)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    W. Heinzer, B. Johnston, D. Lantz, K. Shah, The Ratliff-Rush Ideals in a Noetherian Ring: A Survey, Methods in module theory (Colorado Springs, CO, 1991), Lecture Notes in Pure and Applied Mathematics, vol. 140 (Dekker, New York, 1993) pp. 149–159Google Scholar
  29. 29.
    W.C. Holland, J. Martinez, W.Wm. McGovern M.Tesemma, Bazzoni’s Conjecture. J. Algebra 320, 1764–1768 (2008)Google Scholar
  30. 30.
    J. Lipman, Stable ideals and Arf rings. Amer. J. Math 93, 649–685 (1971)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    E. Matlis, Reflexive domains. J. Algebra 8, 1–33 (1968)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    E. Matlis, Torsion-Free Modules (University of Chicago press, Chicago, 1972)MATHGoogle Scholar
  33. 33.
    A. Mimouni, Ratliff-Rush closure of ideals in integral domains. Glasg. Math. J. 51, 681–689 (2009)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    A. Mimouni, Ratliff-Rush closure of ideals in pullbacks and polynomial rings. Comm. Algebra 37, 3044–3053 (2009)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    A. Mimouni, Pullbacks and Coherent-Like Properties, Advances in commutative ring theory (Fez, 1997), Lecture Notes in Pure and Applied Mathematics, vol. 205 (Dekker, New York, 1999) pp. 437–459Google Scholar
  36. 36.
    W.Wm. McGovern, Prüfer domains with Clifford Class semigroup. J. Commut. Algebra 3, 551–559 (2011)Google Scholar
  37. 37.
    J.L. Mott, M. Zafrullah, On Krull domains. Arch. Math. 56, 559–568 (1991)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    B. Olberding, Globalizing local properties of Prüfer domains. J. Algebra 205, 480–504 (1998)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    B. Olberding, Stability of Ideals and its Applications, Ideal theoretic methods in commutative algebra (Columbia, MO, 1999), Lecture Notes in Pure and Applied Mathematics, vol. 220 (Dekker, New York, 2001) pp. 319–341Google Scholar
  40. 40.
    B. Olberding, Stability, duality and 2-generated ideals, and a canonical decomposition of modules. Rend. Semin. Mat. Univ. Padova 106, 261–290 (2001)MATHMathSciNetGoogle Scholar
  41. 41.
    B. Olberding, On the classification of stable domains. J. Algebra 243, 177–197 (2001)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    B. Olberding, On the structure of stable domains. Comm. Algebra 30, 877–895 (2002)CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    B. Olberding, An Exceptional Class of Stable Domains, Communication at AMS Meeting # 991, (Special Session on Commutative Rings and Monoids, Chapel Hill, NC) October 2003.Google Scholar
  44. 44.
    B. Olberding, Noetherian Rings Without Finite Normalizations, Progress in commutative algebra 2 (Walter de Gruyter, Berlin, 2012) pp. 171–203Google Scholar
  45. 45.
    B. Olberding, One-dimensional bad Noetherian domains, Trans. AMS 366, 4067–4095 (2014)CrossRefMathSciNetGoogle Scholar
  46. 46.
    G. Picozza, F. Tartarone, Flat ideals and stability in integral domains. J. Algebra 324, 1790–1802 (2010)CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    N. Popescu, On a class of Prüfer domains. Rev. Roumanie Math. Pures Appl. 29, 777–786 (1984)MATHGoogle Scholar
  48. 48.
    L.J. Ratliff, Jr., D.E. Rush, Two notes on reductions of ideals. Indiana Univ. Math. J. 27, 929–934 (1978)CrossRefMATHMathSciNetGoogle Scholar
  49. 49.
    D.E. Rush, Rings with two-generated ideals. J. Pure Appl. Algebra 73, 257–275 (1991)CrossRefMATHMathSciNetGoogle Scholar
  50. 50.
    D.E. Rush, Two-generated ideals and representations of abelian groups over valuation rings. J. Algebra 177, 77–101 (1995)CrossRefMATHMathSciNetGoogle Scholar
  51. 51.
    J.D. Sally, W.V. Vasconcelos, Stable rings and a problem of Bass. Bull. AMS 79, 574–576 (1973)CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    J.D. Sally, W.V. Vasconcelos, Stable rings. J. Pure Appl. Algebra 4, 319–336 (1974)CrossRefMATHMathSciNetGoogle Scholar
  53. 53.
    L. Sega, Ideal class semigroups of overrings. J. Algebra 311, 702–713 (2007)CrossRefMATHMathSciNetGoogle Scholar
  54. 54.
    M. Zafrullah, t-Invertibility and Bazzoni-like statements. J. Pure Appl. Algebra 214, 654–657 (2010)Google Scholar
  55. 55.
    P. Zanardo, U. Zannier, The class semigroup of orders in number fields. Math. Proc. Cambridge Philos. Soc. 115, 379–391 (1994)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Dipartimento di Matematica e FisicaUniversità degli Studi Roma TreRomaItaly

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