Ways in which C(X) mod a Prime Ideal Can be a Valuation Domain; Something Old and Something New

  • Bikram Banerjee
  • Melvin Henriksen
Part of the Trends in Mathematics book series (TM)


C(X) denotes the ring of continuous real-valued functions on a Tychonoff space X and P a prime ideal of C(X). We summarize a lot of what is known about the reside class domains C(X)/P and add many new results about this subject with an emphasis on determining when the ordered C(X)/P is a valuation domain (i.e., when given two nonzero elements, one of them must divide the other). The interaction between the space X and the prime ideal P is of great importance in this study. We summarize first what is known when P is a maximal ideal, and then what happens when C(X)/P is a valuation domain for every prime ideal P (in which case X is called an SV-space and C(X) an SV-ring). Two new generalizations are introduced and studied. The first is that of an almost SV-spaces in which each maximal ideal contains a minimal prime ideal P such that C(X)/P is a valuation domain. In the second, we assume that each real maximal ideal that fails to be minimal contains a nonmaximal prime ideal P such that C(X)/P is a valuation domain. Some of our results depend on whether or not βω ω contains a P-point. Some concluding remarks include unsolved problems.


Prime Ideal Maximal Ideal Compact Space Discrete Space Metrizable Space 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [ACCH81]
    M. Antonovskij, D. Chudnovsky, G. Chudnovsky, and E. Hewitt, Rings of real-valued continuous functions. II, Math. Zeit. 176 (1981), 151–186.MATHCrossRefMathSciNetGoogle Scholar
  2. [CD86]
    G. Cherlin and M. Dickmann, Real closed rings I, Fund. Math. 126 (1986), 147–183.MATHMathSciNetGoogle Scholar
  3. [D84]
    A. Dow, On ultrapowers of Boolean algebras, Proceedings of the 1984 topology conference (Auburn, Ala., 1984). Topology Proc. 9 (1984), 269–291.MathSciNetGoogle Scholar
  4. [DGM97]
    J.M. Domínguez, J. Gómez, and M.A. Mulero,. Intermediate algebras between C*(X) and C(X) as rings of fractions of C*(X), Topology & Appl. 77 (1997), 115–130.MATHCrossRefMathSciNetGoogle Scholar
  5. [DW96]
    H.G. Dales and W.H. Woodin, Super-Real Fields, Claredon Press, Oxford, 1994.Google Scholar
  6. [E89]
    R. Engelking, General Topology, Heldermann Verlag Berlin, 1989.MATHGoogle Scholar
  7. [G90]
    L. Gillman, Convex and pseudoprime ideals in C(X), General Topology and Applications, Proceedings of the 1988 Northeast Conference, pp. 87–95, Marcel Dekker Inc., New York 1990.Google Scholar
  8. [GJ60]
    L. Gillman and M. Jerison, Quotient fields of residue class rings of continuous functions, Illinois J. Math 4 (1960), 425–436.MATHMathSciNetGoogle Scholar
  9. [GJ76]
    L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, New York, 1976.MATHGoogle Scholar
  10. [GK60]
    L. Gillman and C.W. Kohls, Convex and pseudoprime ideals in rings of continuous functions, Math. Zeit. 72 (1960), 399–409.MATHCrossRefMathSciNetGoogle Scholar
  11. [Hen97]
    M. Henriksen, Rings of continuous functions in the 1950s. Handbook of the history of general topology, Vol. 1, 243–253, Kluwer Acad. Publ., Dordrecht, 1997.Google Scholar
  12. [Hen02]
    M. Henriksen, Topology related to rings of real-valued continuous functions, Recent Progress in General Topology II, eds. M. Husek, J. van Mill, 553–556, Elsevier Science, 2002.Google Scholar
  13. [Hew48]
    E. Hewitt, Rings of real-valued continuous functions I, Trans. Amer. Math. Soc. 64 (1948) 54–99.CrossRefMathSciNetGoogle Scholar
  14. [HJe65]
    M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110–130.MATHCrossRefMathSciNetGoogle Scholar
  15. [HJo61]
    M. Henriksen and D.G. Johnson, On the structure of a class of archimedean lattice-ordered algebras. Fund. Math 50 (1961), 73–94.MATHMathSciNetGoogle Scholar
  16. [HLMW94]
    M. Henriksen, S. Larson, J. Martinez, and R.G. Woods, Lattice-ordered algebras that are subdirect products of valuation domains. Trans. Amer. Math. Soc. 345 (1994), 195–221.MATHCrossRefMathSciNetGoogle Scholar
  17. [HMW03]
    M. Henriksen, J. Martinez, and R.G. Woods, Spaces X in which every prime z-ideal of C(X) are minimal or maximal, Comment. Math. Univ. Carolinae 44 (2003), 261–294.MATHMathSciNetGoogle Scholar
  18. [HW92a]
    M. Henriksen and R. Wilson, When is C(X)/P a valuation ring for every prime ideal P?. Topology & Appl. 44 (1992), 175–180.MATHCrossRefMathSciNetGoogle Scholar
  19. [HW92b]
    M. Henriksen and R. Wilson, Almost discrete SV-spaces, Topology & Appl. 46 (1992), 89–97.MATHCrossRefMathSciNetGoogle Scholar
  20. [HW04]
    M. Henriksen and R.G. Woods, Cozero complemented spaces; when the space of minimal prime ideals of a C(X) is compact, Topology & Appl. 141 (2004), 147–170.MATHMathSciNetGoogle Scholar
  21. [HIJ62]
    M. Henriksen, J.R. Isbell, and D.G. Johnson, Residue class fields of latticeordered algebras, Fund. Math. 50 1961/1962 107–117.MATHMathSciNetGoogle Scholar
  22. [K58]
    C.W. Kohls, Prime ideals in rings of continuous functions II, Duke Math. J. 25 (1958), 447–458.MATHCrossRefMathSciNetGoogle Scholar
  23. [L86]
    S. Larson, Convexity conditions on f-rings, Canad. J. Math. 38 (1986), 48–64.MATHMathSciNetGoogle Scholar
  24. [L03]
    S. Larson, Constructing rings of continuous functions in which there are many maximal ideals of nontrivial rank, Comm. Alg 31 (2003), 2183–2206.MATHCrossRefMathSciNetGoogle Scholar
  25. [M90]
    J. Maloney, Residue class domains of the ring of convergent sequences and of C([0, 1]), R) and C ([0, 1]), R), Pacific J. Math. 143 (1990), 79–153.MathSciNetGoogle Scholar
  26. [MZ05]
    J. Martinez and E. Zenk, Dimension in algebraic frames II: Applications to frames of ideals in C(X), Comment. Math. Univ. Carolinae 46 (2005) 607–636.MATHMathSciNetGoogle Scholar
  27. [PW88]
    J. Porter and R.G. Woods, Extensions and Absolutes of Hausdorff Spaces, Springer-Verlag, New York 1988.MATHGoogle Scholar
  28. [R82]
    J. Roitman, Nonisomorphic hyper-real fields from nonisomorphic ultrapowers, Math. Zeit. 181 (1982), 93–96.MATHCrossRefMathSciNetGoogle Scholar
  29. [Sc97]
    N. Schwartz, Rings of continuous functions as real closed rings, Ordered algebraic structures (Curaçao, 1995), 277–313, Kluwer Acad. Publ., Dordrecht, 1997.Google Scholar
  30. [Se71]
    Z. Semadeni, Banach Spaces of Continuous Functions, Polish Scientific Publishers, Warsaw 1971.MATHGoogle Scholar
  31. [St36]
    M.H. Stone, Applications of the theory of Boolean rings to general toplogy, Trans. Amer. Math. Soc. 41 (1937), 375–481.MATHCrossRefMathSciNetGoogle Scholar
  32. [Wa74]
    R.C. Walker, The Stone-Čech Compactification, Springer Verlag, New York 1974.Google Scholar
  33. [We75]
    M. Weir, Hewitt-Nachbin Spaces, North-Holland Math. Studies, American Elsevier, New York 1975.Google Scholar
  34. [Wi82]
    E. Wimmers, The Shelah P-point indepence theorem, Israel J. Math. 43 (1982), 28–48.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag AG2007 2007

Authors and Affiliations

  • Bikram Banerjee
    • 1
  • Melvin Henriksen
    • 2
  1. 1.Department.of Pure MathematicsUniversity of CalcuttaWest BengalIndia
  2. 2.Department of MathematicsHarvey Mudd CollegeClaremontUSA

Personalised recommendations