-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.
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.
@article{AFST_2010_6_19_S1_159_0,
author = {Niels Schwartz},
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 = {https://afst.centre-mersenne.org/articles/10.5802/afst.1280/}
}
TY - JOUR AU - Niels Schwartz 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 - https://afst.centre-mersenne.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 Niels Schwartz %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 https://afst.centre-mersenne.org/articles/10.5802/afst.1280/ %R 10.5802/afst.1280 %G en %F AFST_2010_6_19_S1_159_0
Niels Schwartz. $SV$-Rings and $SV$-Porings. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. S1, pp. 159-202. doi : 10.5802/afst.1280. https://afst.centre-mersenne.org/articles/10.5802/afst.1280/
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