Anneaux \`a diviseurs et anneaux de Krull (une approche constructive)

TL;DR

This paper develops a constructive, algorithmic approach to the theory of divisor rings and Krull domains, avoiding reliance on complete factorizations, and provides explicit methods within a Bishop-style framework.

Contribution

It introduces a constructive, algorithmic framework for divisor rings and Krull domains, offering explicit methods without assuming complete factorizations.

Findings

Revisits classical results constructively

Provides explicit algorithms for divisor rings

Avoids assumptions of complete factorizations

Abstract

We give give an elementary and constructive version of the theory of "Pr\"ufer v-Multiplication Domains" (which we call "anneaux \`a diviseurs" in the paper) and Krull Domains. The main results of these theories are revisited from a constructive point of view, following the Bishop style, and without assuming properties of complete factorizations. Nous pr\'esentons dans cet article une approche constructive, dans le style de Bishop, de la th\'eorie des diviseurs et des anneaux de Krull. Nous accordons une place centrale aux "anneaux \`a diviseurs," appel\'es PvMD dans la litt\'erature anglaise. Les r\'esultats classiques sont obtenus comme r\'esultats d'algorithmes explicites sans faire appel aux hypoth\`eses de factorisation compl\`ete.