  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 $1/(1+x^2)$,
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.
