Minimal Prüfer-Dress rings and products of idempotent matrices

Abstract

We investigate a special class of Prüfer domains, firstly introduced by Dress in 1965. The minimal Dress ring DK , of a field K, is the smallest subring of K that contains every element of the form 1/(1 + x 2 ), with x ∈ K. We show that, for some choices of K, DK may be a valuation domain, or, more generally, a Bézout domain admitting a weak algorithm. Then we focus on the minimal Dress ring D of R(X): we describe its elements, we prove that it is a Dedekind domain and we characterize its non-principal ideals. Moreover, we study the products of 2 × 2 idempotent matrices over D, a subject of particular interest for Prüfer non-Bézout domains.