Skip to main content

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

  • Chapter

Part of the Trends in Mathematics book series (TM)

Abstract

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.

Keywords

  • 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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-7643-8478-4_1
  • Chapter length: 25 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   129.00
Price excludes VAT (USA)
  • ISBN: 978-3-7643-8478-4
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book
USD   169.00
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Antonovskij, D. Chudnovsky, G. Chudnovsky, and E. Hewitt, Rings of real-valued continuous functions. II, Math. Zeit. 176 (1981), 151–186.

    MATH  CrossRef  MathSciNet  Google Scholar 

  2. G. Cherlin and M. Dickmann, Real closed rings I, Fund. Math. 126 (1986), 147–183.

    MATH  MathSciNet  Google Scholar 

  3. A. Dow, On ultrapowers of Boolean algebras, Proceedings of the 1984 topology conference (Auburn, Ala., 1984). Topology Proc. 9 (1984), 269–291.

    MathSciNet  Google Scholar 

  4. 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.

    MATH  CrossRef  MathSciNet  Google Scholar 

  5. H.G. Dales and W.H. Woodin, Super-Real Fields, Claredon Press, Oxford, 1994.

    Google Scholar 

  6. R. Engelking, General Topology, Heldermann Verlag Berlin, 1989.

    MATH  Google Scholar 

  7. 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. L. Gillman and M. Jerison, Quotient fields of residue class rings of continuous functions, Illinois J. Math 4 (1960), 425–436.

    MATH  MathSciNet  Google Scholar 

  9. L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, New York, 1976.

    MATH  Google Scholar 

  10. L. Gillman and C.W. Kohls, Convex and pseudoprime ideals in rings of continuous functions, Math. Zeit. 72 (1960), 399–409.

    MATH  CrossRef  MathSciNet  Google Scholar 

  11. 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. 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. E. Hewitt, Rings of real-valued continuous functions I, Trans. Amer. Math. Soc. 64 (1948) 54–99.

    CrossRef  MathSciNet  Google Scholar 

  14. M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110–130.

    MATH  CrossRef  MathSciNet  Google Scholar 

  15. M. Henriksen and D.G. Johnson, On the structure of a class of archimedean lattice-ordered algebras. Fund. Math 50 (1961), 73–94.

    MATH  MathSciNet  Google Scholar 

  16. 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.

    MATH  CrossRef  MathSciNet  Google Scholar 

  17. 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.

    MATH  MathSciNet  Google Scholar 

  18. M. Henriksen and R. Wilson, When is C(X)/P a valuation ring for every prime ideal P?. Topology & Appl. 44 (1992), 175–180.

    MATH  CrossRef  MathSciNet  Google Scholar 

  19. M. Henriksen and R. Wilson, Almost discrete SV-spaces, Topology & Appl. 46 (1992), 89–97.

    MATH  CrossRef  MathSciNet  Google Scholar 

  20. 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.

    MATH  MathSciNet  Google Scholar 

  21. M. Henriksen, J.R. Isbell, and D.G. Johnson, Residue class fields of latticeordered algebras, Fund. Math. 50 1961/1962 107–117.

    MATH  MathSciNet  Google Scholar 

  22. C.W. Kohls, Prime ideals in rings of continuous functions II, Duke Math. J. 25 (1958), 447–458.

    MATH  CrossRef  MathSciNet  Google Scholar 

  23. S. Larson, Convexity conditions on f-rings, Canad. J. Math. 38 (1986), 48–64.

    MATH  MathSciNet  Google Scholar 

  24. S. Larson, Constructing rings of continuous functions in which there are many maximal ideals of nontrivial rank, Comm. Alg 31 (2003), 2183–2206.

    MATH  CrossRef  MathSciNet  Google Scholar 

  25. 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.

    MathSciNet  Google Scholar 

  26. 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.

    MATH  MathSciNet  Google Scholar 

  27. J. Porter and R.G. Woods, Extensions and Absolutes of Hausdorff Spaces, Springer-Verlag, New York 1988.

    MATH  Google Scholar 

  28. J. Roitman, Nonisomorphic hyper-real fields from nonisomorphic ultrapowers, Math. Zeit. 181 (1982), 93–96.

    MATH  CrossRef  MathSciNet  Google Scholar 

  29. 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. Z. Semadeni, Banach Spaces of Continuous Functions, Polish Scientific Publishers, Warsaw 1971.

    MATH  Google Scholar 

  31. M.H. Stone, Applications of the theory of Boolean rings to general toplogy, Trans. Amer. Math. Soc. 41 (1937), 375–481.

    MATH  CrossRef  MathSciNet  Google Scholar 

  32. R.C. Walker, The Stone-Čech Compactification, Springer Verlag, New York 1974.

    Google Scholar 

  33. M. Weir, Hewitt-Nachbin Spaces, North-Holland Math. Studies, American Elsevier, New York 1975.

    Google Scholar 

  34. E. Wimmers, The Shelah P-point indepence theorem, Israel J. Math. 43 (1982), 28–48.

    MATH  CrossRef  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2007 Birkhäuser Verlag AG2007

About this chapter

Cite this chapter

Banerjee, B., Henriksen, M. (2007). Ways in which C(X) mod a Prime Ideal Can be a Valuation Domain; Something Old and Something New. In: Boulabiar, K., Buskes, G., Triki, A. (eds) Positivity. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8478-4_1

Download citation