$SV$-Rings and $SV$-Porings

Schwartz, Niels¹

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

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Numéro spécial : Actes de la Conférence sur les anneaux ordonnés en honneur de Melvin Henriksen, Université d’état de Louisiane à Bâton Rouge, 2007, Tome 19 (2010) no. S1, pp. 159-202.

Abstract (VO)

$SV$-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.

Résumé

Les $SV$-anneaux sont les anneaux commutatifs dont les quotients modulo leurs idéaux premiers sont des anneaux de valuation. Les $SV$-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 $SV$-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 $SV$-anneaux partiellement ordonnés ($SV$-porings) et nous démontrons quelques propriétés élémentaires de ces anneaux. Finalement, $SV$-anneaux sont utilisés pour étudier les sous-anneaux convexes et les extensions convexes des anneaux partiellement ordonnés et, en particulier, des anneaux réels clos.

Détail

DOI : 10.5802/afst.1280

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