Les -anneaux sont les anneaux commutatifs dont les quotients modulo leurs idéaux premiers sont des anneaux de valuation. Les -anneaux apparaissent de la façon la plus naturelle en connexion avec les anneaux partiellement ordonnés (= porings) ; ils ont été étudiés uniquement dans ce contexte jusqu’à présent. Dans le présent article, pour la première fois nous developpons la théorie des -anneaux d’une manière systématique, sans supposer la présence d’un ordre partiel. Une attention particulière est consacrée à la question d’axiomatisabilité (au sens de la théorie des modèles). Nous introduisons les -anneaux partiellement ordonnés (-porings) et nous démontrons quelques propriétés élémentaires de ces anneaux. Finalement, -anneaux sont utilisés pour étudier les sous-anneaux convexes et les extensions convexes des anneaux partiellement ordonnés et, en particulier, des anneaux réels clos.
-rings are commutative rings whose factor rings modulo prime ideals are valuation rings. -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 -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 -rings (-porings) are introduced, and some elementary properties are exhibited. Finally, -rings are used to study convex subrings and convex extensions of porings, in particular of real closed rings.
@article{AFST_2010_6_19_S1_159_0,
author = {Schwartz, Niels},
title = {$SV${-Rings} and $SV${-Porings}},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {159--202},
publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 19},
number = {S1},
year = {2010},
doi = {10.5802/afst.1280},
mrnumber = {2675726},
language = {en},
url = {http://www.numdam.org/articles/10.5802/afst.1280/}
}
TY - JOUR AU - Schwartz, Niels TI - $SV$-Rings and $SV$-Porings JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2010 SP - 159 EP - 202 VL - 19 IS - S1 PB - Université Paul Sabatier, Institut de mathématiques PP - Toulouse UR - http://www.numdam.org/articles/10.5802/afst.1280/ DO - 10.5802/afst.1280 LA - en ID - AFST_2010_6_19_S1_159_0 ER -
%0 Journal Article %A Schwartz, Niels %T $SV$-Rings and $SV$-Porings %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2010 %P 159-202 %V 19 %N S1 %I Université Paul Sabatier, Institut de mathématiques %C Toulouse %U http://www.numdam.org/articles/10.5802/afst.1280/ %R 10.5802/afst.1280 %G en %F AFST_2010_6_19_S1_159_0
Schwartz, Niels. $SV$-Rings and $SV$-Porings. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. S1, pp. 159-202. doi : 10.5802/afst.1280. http://www.numdam.org/articles/10.5802/afst.1280/
[1] Brumfiel (G.W.).— Partially Ordered Rings and Semi-Algebraic Geometry. London Math. Soc. Lecture Note Series, vol. 37, Camb. Univ. Press, Cambridge (1979).
[2] Carral (M.), Coste (M.).— Normal spectral spaces and their dimensions. J. Pure Applied Alg. 30, p. 227-235 (1983).
[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).
[5] Cherlin (G.), Dickmann (M.A.).— Real closed rings I. Residue rings of rings of continuous functions. Fund. Math. 126, p. 147-183 (1986).
[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).
[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).
[9] Gilmer (R.).— Multiplicative Ideal Theory. Marcel Dekker, New York (1972).
[10] Glaz (S.).— Commutative Coherent Rings. Lecture Notes in Maths., vol. 1371, Springer-Verlag, Berlin (1989).
[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).
[13] Henriksen (M.), Larson (S.).— Semiprime -rings that are subdirect products of valuation domains. In: Ordered Algebraic Structures (Eds. J. Martinez, W.C. Holland), Kluwer, Dordrecht, p. 159-168 (1993).
[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).
[15] Henriksen (M.), Wilson.— When is a valuation ring for every prime ideal P? Topology and its Applications 44, p. 175-180 (1992).
[16] Henriksen (M.), Wilson.— Almost discrete -spaces. Topology and its Applications 46, p. 89-97 (1992).
[17] Hochster (M.).— Prime ideal structure in commutative rings. Trans Amer. Math. Soc. 142, p. 43-60 (1969).
[18] Hodges (W.).— Model Theory. Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge Univ. Press, Cambridge (1993).
[19] Johnstone (P.).— Stone Spaces. Cambridge Univ. Press, Cambridge (1982).
[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 -rings. Can. J. Math. 38, p. 48-64 (1986).
[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).
[25] Larson (S.).— Images and Open Subspaces of -Spaces. Preprint 2007.
[26] Matsumura (H.).— Commutative Algebra. 2nd Ed. Benjamin/Cummings, Reading, Massachusetts 1980.
[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).
[28] Sch¨ulting (H.W.).— On real places of a field and their holomorphy ring. Comm. Alg. 10, p. 1239-1284 (1982).
[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).
[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).
[33] Schwartz (N.).— Epimorphic extensions and Prüfer extensions of partially ordered rings. Manuscripta mathematica 102, p. 347-381 (2000).
[34] Schwartz (N.).— Convex subrings of partially ordered rings. To appear: Math. Nachr.
[35] Schwartz (N.).— Real closed valuation rings. Comm. in Alg. 37, p. 3796-3814 (2009).
[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).
[37] Schwartz (N.), Tressl (M.).— Elementary properties of minimal and maximal points in Zariski spectra., Journal of Algebra 323, p. 698-728 (2010).
[38] Walker (R.C.).— The Stone-Cech Compactification. Springer, Berlin (1974).
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