Localizing Systems and Semistar Operations

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

Abstract

In 1994 A. Okabe and R. Matsuda [22] introduced the notion of semistar operation; see also, [21] and [20]. This concept extends the classical concept of star operation, as developed in Gilmer’s book [12], and hence the related classical theory of ideal systems based on the works of W. Krull, E. Noether, H. Prüfer, and P. Lorenzen from the 1930’s. For a systematic treatment of these ideas, see the books by P. Jaffard [17] and F. Halter-Koch [14], where a complete and updated bibliography is available.

Keywords

Prime Ideal Integral Domain Finite Type Integral Ideal Valuation 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. [1]
    D. D. Anderson, Star operations induced by overrings, Comm. Algebra 16 (1988), 2535–2553.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    D. D. Anderson and D. F. Anderson, Some remarks on star operations and the class group, J. Pure Appl. Algebra 51 (1988), 27–33.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    D. D. Anderson and S. J. Cook, Star operations and their induced lattices, preprint.Google Scholar
  4. [4]
    N. Bourbaki, Algèbre Commutative, Hermann, Paris, 1961–1965.MATHGoogle Scholar
  5. [5]
    V. Barucci, D. Dobbs, and M. Fontana, Conducive integral domains as pullbacks, Manuscripta Math. 54 (1986), 261–277.MathSciNetCrossRefGoogle Scholar
  6. [6]
    E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form “D + M”, Michigan Math. J. 20 (1973), 79–95.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Wang Fanggui and R.L. McCasland, On w-modules over strong Mori domains, Comm. Algebra 25 (1997), 1285–1306.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    M. Fontana, Kaplansky ideal transform: a survey, M. Dekker Lect. Notes 205 (1999), 271–306.MathSciNetGoogle Scholar
  9. [9]
    M. Fontana, J. Huckaba, and I. Papick, Prüfer Domains, M. Dekker, New York, 1997.MATHGoogle Scholar
  10. [10]
    S. Gabelli, Prüfer (##) domains and localizing systems of ideals, M. Dekker Lect. Notes, 205 (1999), 391–410.MathSciNetGoogle Scholar
  11. [11]
    J. Garcia, P. Jara, and E. Santos, Prüfer * -multiplication domains and torsion theories, Comm. Algebra, 27 (1999), 1275–1295.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    R. Gilmer, Multiplicative Ideal Theory, M. Dekker, New York, 1972.MATHGoogle Scholar
  13. [13]
    F. Halter-Koch, Kronecker function rings and generalized integral closures, preprint 1999.Google Scholar
  14. [14]
    F. Halter-Koch, Ideal Systems: An Introduction to Multiplicative Ideal Theory, M. Dekker, New York, 1998.Google Scholar
  15. [15]
    W. Heinzer and J. Ohm, An essential ring which is not a v-multiplication ring, Can. J. Math., 25 (1973), 856–861.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    J.M. Garcia Hernandez, Radicales de anillos y modulos noetherianos relativos, Ph.D. Thesis, Univ. Granada (1995).Google Scholar
  17. [17]
    P. Jaffard, Les Systèmes d’Idéaux, Dunod, Paris, 1960.MATHGoogle Scholar
  18. [18]
    B. Kang, *-Operations On Integral Domains, Ph.D. Dissertation, The University of Iowa, 1987.Google Scholar
  19. [19]
    R. Matsuda, Kronecker function rings of semistar operations on rings, Algebra Colloquium, 5 (1998), 241–254.MathSciNetMATHGoogle Scholar
  20. [20]
    R. Matsuda and I. Sato, Note on star-operations and semistar operations, Bull. Fac. Sci. Ibaraki Univ., 28 (1996), 155–161.CrossRefGoogle Scholar
  21. [21]
    R. Matsuda and T. Sugatani, Semistar operations on integral domains, II. Math. J. Toyama Univ., 18 (1995), 155–161.MathSciNetMATHGoogle Scholar
  22. [22]
    A. Okabe and R. Matsuda, Semistar-operations on integral domains, Math. J. Toyama Univ. 17 (1994), 1–21.MathSciNetMATHGoogle Scholar
  23. [23]
    A. Okabe and R. Matsuda, Kronecker function rings of semistar operations, Tsukuba J. Math., 21 (1997), 529–548.MathSciNetMATHGoogle Scholar
  24. [24]
    N. Popescu, A characterization of generalized Dedekind domains, Rev. Roumaine Math. Pures Appl. 29 (1984), 777–786.MathSciNetMATHGoogle Scholar
  25. [25]
    B. Stenström, Rings of Quotients, Springer, Berlin 1975.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Marco Fontana
    • 1
  • James A. Huckaba
    • 2
  1. 1.Dipartimento di MatematicaUniversità degli Studi Roma TreRomeItaly
  2. 2.Department of MathematicsUniversity of Missouri-ColumbiaColumbiaUSA

Personalised recommendations