Vol. 1, No. 3, 2007

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Sur la définissabilité existentielle de la non-nullité dans les anneaux

Laurent Moret-Bailly

Vol. 1 (2007), No. 3, 331–346
Abstract

On étudie les anneaux (notamment noethériens) dans lesquels l’ensemble des éléments non nuls est existentiel positif (réunion finie de projections d’ensembles « algébriques »). Dans le cas noethérien intègre, on montre notamment que cette condition est vérifiée pour tout anneau qui n’est pas local hensélien, et qu’elle ne l’est pas pour un anneau local hensélien excellent qui n’est pas un corps.

Ces résultats apportent au passage une réponse à une question de Popescu sur l’approximation forte pour les couples henséliens.

We investigate rings in which the set of nonzero elements is positive-existential (i.e., a finite union of projections of “algebraic” sets). In the case of Noetherian domains, we prove in particular that this condition is satisfied whenever the ring in question is not local Henselian, while it is not satisfied for any excellent local Henselian domain which is not a field.

As a byproduct, we obtain an answer to a question of Popescu on strong approximation for Henselian pairs.

Keywords
Noetherian rings, positive-existential definability, Henselian rings, Artin approximation
Mathematical Subject Classification 2000
Primary: 13E05
Secondary: 03C99, 11U09, 13B40, 13J15
Milestones
Received: 30 July 2007
Revised: 18 September 2007
Accepted: 18 October 2007
Published: 1 August 2007
Authors
Laurent Moret-Bailly
IRMAR (Institut de Recherche Mathématique de Rennes, UMR 6625 du CNRS)
Université de Rennes 1
Campus de Beaulieu
F-35042 Rennes Cedex
France
http://perso.univ-rennes1.fr/laurent.moret-bailly/