Positivity pp 1-25 | Cite as

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

## 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## Preview

Unable to display preview. Download preview PDF.

## References

- [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 - [CD86]G. Cherlin and M. Dickmann,
*Real closed rings I*, Fund. Math. 126 (1986), 147–183.MATHMathSciNetGoogle Scholar - [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 - [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 - [DW96]H.G. Dales and W.H. Woodin,
*Super-Real Fields*, Claredon Press, Oxford, 1994.Google Scholar - [E89]R. Engelking,
*General Topology*, Heldermann Verlag Berlin, 1989.MATHGoogle Scholar - [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 - [GJ60]L. Gillman and M. Jerison,
*Quotient fields of residue class rings of continuous functions*, Illinois J. Math 4 (1960), 425–436.MATHMathSciNetGoogle Scholar - [GJ76]L. Gillman and M. Jerison,
*Rings of Continuous Functions*, Springer-Verlag, New York, 1976.MATHGoogle Scholar - [GK60]L. Gillman and C.W. Kohls,
*Convex and pseudoprime ideals in rings of continuous functions*, Math. Zeit. 72 (1960), 399–409.MATHCrossRefMathSciNetGoogle Scholar - [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 - [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 - [Hew48]E. Hewitt,
*Rings of real-valued continuous functions I*, Trans. Amer. Math. Soc. 64 (1948) 54–99.CrossRefMathSciNetGoogle Scholar - [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 - [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 - [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 - [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 - [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 - [HW92b]M. Henriksen and R. Wilson,
*Almost discrete SV-spaces*, Topology & Appl. 46 (1992), 89–97.MATHCrossRefMathSciNetGoogle Scholar - [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 - [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 - [K58]C.W. Kohls,
*Prime ideals in rings of continuous functions II*, Duke Math. J. 25 (1958), 447–458.MATHCrossRefMathSciNetGoogle Scholar - [L86]S. Larson,
*Convexity conditions on f-rings*, Canad. J. Math. 38 (1986), 48–64.MATHMathSciNetGoogle Scholar - [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 - [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 - [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 - [PW88]J. Porter and R.G. Woods,
*Extensions and Absolutes of Hausdorff Spaces*, Springer-Verlag, New York 1988.MATHGoogle Scholar - [R82]J. Roitman,
*Nonisomorphic hyper-real fields from nonisomorphic ultrapowers*, Math. Zeit. 181 (1982), 93–96.MATHCrossRefMathSciNetGoogle Scholar - [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 - [Se71]Z. Semadeni,
*Banach Spaces of Continuous Functions*, Polish Scientific Publishers, Warsaw 1971.MATHGoogle Scholar - [St36]M.H. Stone,
*Applications of the theory of Boolean rings to general toplogy*, Trans. Amer. Math. Soc. 41 (1937), 375–481.MATHCrossRefMathSciNetGoogle Scholar - [Wa74]R.C. Walker,
*The Stone-Čech Compactification*, Springer Verlag, New York 1974.Google Scholar - [We75]M. Weir,
*Hewitt-Nachbin Spaces*, North-Holland Math. Studies, American Elsevier, New York 1975.Google Scholar - [Wi82]E. Wimmers,
*The Shelah P-point indepence theorem*, Israel J. Math. 43 (1982), 28–48.MATHCrossRefMathSciNetGoogle Scholar