Constructive Theory of Pruefer Rings
by
Henri Lombardi
Universite de Franche-Comte, Besancon

We study by elementary and constructive methods the basic theory of Prüfer rings. We adopt the most popular definition in the case where zero divisors are allowed: a Prüfer ring is a ring for which any finitely generated ideal is flat.

In classical proofs, we deal with localizations at any maximal ideal, getting valuation rings. In order to get constructive proofs we use a close inspection of the classical proof for the case of valuation rings. We see that the proof involves some finite computations under the hypothesis: any element is in the maximal ideal or is invertible. The constructive rereading consists in considering localisations for which any relevant element is in the radical (of the localized ring) or is invertible (in the localized ring). Instead of localizations at maximal ideals we use well controlled localizations, at multiplicatively closed subsets S_i that are described in finite terms, the corresponding U_Si being an open covering of the Zariski spectrum.

We think that we are showing in practice that many classical proofs are in fact constructive.

Date received: May 26, 1999