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

Abstract

We investigate the 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.

arXiv comment16 pages, in French. Accepted for publication in Algebra and Number Theory. Change of title and minor changes in the text, following referee's suggestions. Report number corrected, hyperlinks added