On the Flatness of the Epimorphic Hull of a Ring of Continuous Functions

  • R. Raphael
  • R. G. Woods
Part of the Developments in Mathematics book series (DEVM, volume 7)


For commutative semiprime rings R, the classical ring of quotients Q ci (R) is R-flat, but the epimorphic hull E(R) need not be. An example due to Quentel shows that E(R) can be flat and still not coincide with Q ci (R). In Proposition 7 below we show that such behaviour is excluded for rings of the form C(X). A related question is addressed, and we characterize, for any cardinal α, the Tychonoff spaces X for which all ideals of C(X) are essentially α-generated.


Manifold Hull Topo Versalis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [B]
    N. Bourbaki, Eléments de mathématique. Fasc. XXVII: Algèbre commutative; chapitre 1: Modules plats; chapitre 2: Localisation; (1961) Hermann, Paris.Google Scholar
  2. [C]
    V. Cateforis, Flat regular quotient rings. Trans. AMS 138 (1969), 241–249.MathSciNetMATHCrossRefGoogle Scholar
  3. [G]
    L. Gillman, Countable generated ideals in rings of continuous functions. Proc. AMS 11 (1960), 660–666.MathSciNetMATHCrossRefGoogle Scholar
  4. [GJ]
    L. Gillman and M. Jerison, Rings of Continuous Functions. (1960) Van Nostrand, Princeton.MATHGoogle Scholar
  5. [HM1]
    A. W. Hager and J. Martinez, Fraction dense algebras and spaces. Can. J. Math, 45 (5) (1993), 977–996.MathSciNetMATHCrossRefGoogle Scholar
  6. [HM2]
    A. W. Hager and J. Martinez, The ring of a-quotients. To appear; Alg. Universalis.Google Scholar
  7. [HJ]
    M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring. Trans. AMS 115 (1965), 110–130.MathSciNetMATHCrossRefGoogle Scholar
  8. [H]
    R. Hodel, Cardinal Functions, I. Handbook of Set-Theoretic Topology (1984) North-Holland, Amsterdam, 1–61.Google Scholar
  9. [J]
    I. Juhasz, Cardinal Functions in Topology. Math. Centrum Tracts 34 (1971) Amsterdam.MATHGoogle Scholar
  10. [Ke]
    J. F. Kennison, Structure and costructure for strongly regular rings. J. Pure and Appl. Algebra 5 (1974) 321–332.MathSciNetMATHCrossRefGoogle Scholar
  11. [L]
    J. Lambek, Lectures on Rings and Modules. (1976) Chelsea Publ. Co., New York.MATHGoogle Scholar
  12. [M]
    J. Martinez, The maximal ring of quotients of an f-ring. Alg. Universalis 33 (1995), 355–369.MATHCrossRefGoogle Scholar
  13. [O1]
    J. P. Olivier, Anneaux absolument plats universels et epimorphismes d’anneaux. C. R. Acad. Sci. Paris 266 Serie A (5 fevrier 1968), 317–318.MathSciNetMATHGoogle Scholar
  14. [O2]
    J. P. Olivier, L’anneau absolument plat universel, les epimorphismes et les parties constructibles. Bol. de la Soc. Mat. Mexicana 23 (1978) 68–74.MathSciNetMATHGoogle Scholar
  15. [Q]
    Y. Quentel, Sur la compacité du spectre minimal d’un anneau. Bull. Soc. Math. France 99 (1971), 265–272.MathSciNetMATHGoogle Scholar
  16. [R]
    R. Raphael, Injective rings. Comm. in Algebra 1 (1974), 403–414.MathSciNetMATHCrossRefGoogle Scholar
  17. [S]
    H. H. Storrer, Epimorphismen von Kommutativen Ringen. Comm. Math. Helv. 43 (1968), 378–401.MathSciNetMATHCrossRefGoogle Scholar
  18. [W]
    R. Wiegand, Modules over universal regular rings. Pac. J. Math. 39 (1971), 807–819.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • R. Raphael
    • 1
  • R. G. Woods
    • 2
  1. 1.Department of MathematicsConcordia UniversityMontrealCanada
  2. 2.Department of MathematicsUniversity of ManitobaWinnipegCanada

Personalised recommendations