S V -Rings and S V -Porings

Niels Schwartz[1]

  • [1] Fakultät für Informatik und Mathematik, Universität Passau, Postfach 2540, 94030 Passau, Germany

Annales de la faculté des sciences de Toulouse Mathématiques (2010)

  • Volume: 19, Issue: S1, page 159-202
  • ISSN: 0240-2963

Abstract

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S V -rings are commutative rings whose factor rings modulo prime ideals are valuation rings. S V -rings occur most naturally in connection with partially ordered rings (= porings) and have been studied only in this context so far. The present note first develops the theory of S V -rings systematically, without assuming the presence of a partial order. Particular attention is paid to the question of axiomatizability (in the sense of model theory). Partially ordered S V -rings ( S V -porings) are introduced, and some elementary properties are exhibited. Finally, S V -rings are used to study convex subrings and convex extensions of porings, in particular of real closed rings.

How to cite

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Schwartz, Niels. "$SV$-Rings and $SV$-Porings." Annales de la faculté des sciences de Toulouse Mathématiques 19.S1 (2010): 159-202. <http://eudml.org/doc/115895>.

@article{Schwartz2010,
abstract = {$SV$-rings are commutative rings whose factor rings modulo prime ideals are valuation rings. $SV$-rings occur most naturally in connection with partially ordered rings (= porings) and have been studied only in this context so far. The present note first develops the theory of $SV$-rings systematically, without assuming the presence of a partial order. Particular attention is paid to the question of axiomatizability (in the sense of model theory). Partially ordered $SV$-rings ($SV$-porings) are introduced, and some elementary properties are exhibited. Finally, $SV$-rings are used to study convex subrings and convex extensions of porings, in particular of real closed rings.},
affiliation = {Fakultät für Informatik und Mathematik, Universität Passau, Postfach 2540, 94030 Passau, Germany},
author = {Schwartz, Niels},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {SV-rings; axiomatizability; convex subrings; real closed rings; partially ordered rings},
language = {eng},
month = {4},
number = {S1},
pages = {159-202},
publisher = {Université Paul Sabatier, Toulouse},
title = {$SV$-Rings and $SV$-Porings},
url = {http://eudml.org/doc/115895},
volume = {19},
year = {2010},
}

TY - JOUR
AU - Schwartz, Niels
TI - $SV$-Rings and $SV$-Porings
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2010/4//
PB - Université Paul Sabatier, Toulouse
VL - 19
IS - S1
SP - 159
EP - 202
AB - $SV$-rings are commutative rings whose factor rings modulo prime ideals are valuation rings. $SV$-rings occur most naturally in connection with partially ordered rings (= porings) and have been studied only in this context so far. The present note first develops the theory of $SV$-rings systematically, without assuming the presence of a partial order. Particular attention is paid to the question of axiomatizability (in the sense of model theory). Partially ordered $SV$-rings ($SV$-porings) are introduced, and some elementary properties are exhibited. Finally, $SV$-rings are used to study convex subrings and convex extensions of porings, in particular of real closed rings.
LA - eng
KW - SV-rings; axiomatizability; convex subrings; real closed rings; partially ordered rings
UR - http://eudml.org/doc/115895
ER -

References

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  1. Brumfiel (G.W.).— Partially Ordered Rings and Semi-Algebraic Geometry. London Math. Soc. Lecture Note Series, vol. 37, Camb. Univ. Press, Cambridge (1979). Zbl0415.13015
  2. Carral (M.), Coste (M.).— Normal spectral spaces and their dimensions. J. Pure Applied Alg. 30, p. 227-235 (1983). Zbl0525.14015
  3. Chang (C.C.), Keisler (H.J.).— Model Theory. 2nd edition. North Holland, Amsterdam (1977). 
  4. Cherlin (G.), Dickmann (M.A.).— Real closed rings II. Model theory. Annals Pure Applied Logic 25, p. 213-231 (1983). Zbl0538.03028
  5. Cherlin (G.), Dickmann (M.A.).— Real closed rings I. Residue rings of rings of continuous functions. Fund. Math. 126, p. 147-183 (1986). Zbl0605.54014
  6. Delfs (H.), Knebusch (M.).— Semialgebraic Topology over a Real Closed Field II – Basic Theory of Semialgebraic Spaces. Math. Z. 178, p. 175-213 (1981). Zbl0447.14003
  7. Engler (A.J.), Prestel (A.).— Valued Fields. Springer-Verlag, Berlin Heidelberg (2005). 
  8. Gillman (L.), Jerison (M.).— Rings of Continuous Functions. Graduate Texts in Maths., Vol. 43, Springer, Berlin (1976). Zbl0339.46018
  9. Gilmer (R.).— Multiplicative Ideal Theory. Marcel Dekker, New York (1972). Zbl0248.13001
  10. Glaz (S.).— Commutative Coherent Rings. Lecture Notes in Maths., vol. 1371, Springer-Verlag, Berlin (1989). Zbl0745.13004
  11. Henriksen (M.), Isbell (J.R.), Johnson (D.G.).— Residue class fields of latticeordered algebras. Fund. Math. 50, p. 73-94 (1961). 
  12. Henriksen (M.), Jerison (M.).— The space of minimal prime ideals of a commutative ring. Trans. AMS 115, p. 110-130 (1965). Zbl0147.29105
  13. Henriksen (M.), Larson (S.).— Semiprime f -rings that are subdirect products of valuation domains. In: Ordered Algebraic Structures (Eds. J. Martinez, W.C. Holland), Kluwer, Dordrecht, p. 159-168 (1993). Zbl0799.06031
  14. Henriksen (M.), Larson (S.), Martinez (J.), Woods (G.R.).— Lattice-ordered algebras that are subdirect products of valuation domains. Trans. Amer. Math. Soc. 345, p. 193-221 (1994). Zbl0817.06014
  15. Henriksen (M.), Wilson.— When is C ( X ) / P a valuation ring for every prime ideal P? Topology and its Applications 44, p. 175-180 (1992). Zbl0801.54014
  16. Henriksen (M.), Wilson.— Almost discrete S V -spaces. Topology and its Applications 46, p. 89-97 (1992). Zbl0791.54049
  17. Hochster (M.).— Prime ideal structure in commutative rings. Trans Amer. Math. Soc. 142, p. 43-60 (1969). Zbl0184.29401
  18. Hodges (W.).— Model Theory. Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge Univ. Press, Cambridge (1993). Zbl0789.03031
  19. Johnstone (P.).— Stone Spaces. Cambridge Univ. Press, Cambridge (1982). Zbl0499.54001
  20. Knebusch (M.), Scheiderer (C.).— Einführung in die reelle Algebra. Vieweg, Braunschweig (1989). 
  21. Knebusch (M.), Zhang (D.).— Manis valuations and Prüferextensions I – A new chapter in commutative algebra. Lecture Notes in Math., vol 1791, Springer, Berlin (2002). 
  22. Knebusch (M.), Zhang (D.).— Convexity, Valuations and PrüferExtensions in Real Algebra. Documenta Math. 10, p. 1-109 (2005). 
  23. Larson (S.).— Convexity conditions on f -rings. Can. J. Math. 38, p. 48-64 (1986). Zbl0588.06011
  24. Larson (S.).— Constructing rings of continuous functions in which there are many maximal ideals with nontrivial rank. Comm. Alg. 31, p. 2183-2206 (2003). Zbl1024.54015
  25. Larson (S.).— Images and Open Subspaces of S V -Spaces. Preprint 2007. 
  26. Matsumura (H.).— Commutative Algebra. 2nd Ed. Benjamin/Cummings, Reading, Massachusetts 1980. Zbl0441.13001
  27. Prestel (A.), Schwartz (N.).— Model theory of real closed rings. In: Fields Institute Communications, vol. 32 (Eds. F.-V. Kuhlmann et al.), American Mathematical Society, Providence, p. 261-290 ( 2002). Zbl1018.12009
  28. Sch¨ulting (H.W.).— On real places of a field and their holomorphy ring. Comm. Alg. 10, p. 1239-1284 (1982). Zbl0509.14026
  29. Schwartz (N.).— Real Closed Rings. In: Algebra and Order (Ed. S. Wolfenstein), Heldermann Verlag, Berlin, p. 175-194 (1986). 
  30. Schwartz (N.).— The basic theory of real closed spaces. Memoirs Amer. Math. Soc. No. 397, Amer. Math. Soc., Providence (1989). 
  31. Schwartz (N.).— Eine universelle Eigenschaft reell abgeschlossener Räume. Comm. Alg. 18, p. 755-774 (1990). Zbl0726.14036
  32. Schwartz (N.).— Rings of continuous functions as real closed rings. In: Ordered Algebraic Structures (Eds. W.C. Holland, J. Martinez), Kluwer, Dordrecht, p. 277-313 (1997). Zbl0885.46024
  33. Schwartz (N.).— Epimorphic extensions and Prüfer extensions of partially ordered rings. Manuscripta mathematica 102, p. 347-381 (2000). Zbl0966.13018
  34. Schwartz (N.).— Convex subrings of partially ordered rings. To appear: Math. Nachr. Zbl1196.13015
  35. Schwartz (N.).— Real closed valuation rings. Comm. in Alg. 37, p. 3796-3814 (2009). Zbl1184.13072
  36. Schwartz (N.), Madden (J.J.).— Semi-algebraic Function Rings and Reflectors of Partially Ordered Rings. Lecture Notes in Mathematics, Vol. 1712, Springer-Verlag, Berlin (1999). Zbl0967.14038
  37. Schwartz (N.), Tressl (M.).— Elementary properties of minimal and maximal points in Zariski spectra., Journal of Algebra 323, p. 698-728 (2010). Zbl1198.13023
  38. Walker (R.C.).— The Stone-Cech Compactification. Springer, Berlin (1974). Zbl0292.54001

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