Minimal Pr\"ufer-Dress rings and products of idempotent matrices

We investigate a special class of Pr"ufer domains, firstly introduced by Dress in 1965. The {\it minimal Dress ring} $D_K$, of a field $K$, is the smallest subring of $K$ that contains every element of the form $1\mathrm{/}(1+x^2)$, with $x\in K$. We show that, for some choices of $K$, $D_K$ may be a valuation domain, or, more generally, a B'ezout domain admitting a weak algorithm. Then we focus on the minimal Dress ring $D$ of $\mathbb{R}(X)$: 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 $2\times 2$ idempotent matrices over $D$, a subject of particular interest for Pr"ufer non-B'ezout domains.