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

TL;DR

This paper explores the structure of minimal Dress rings, especially over real rational functions, showing they can be valuation or Bézout domains, and investigates products of idempotent matrices over these rings.

Contribution

It characterizes the minimal Dress ring of alculus over alculus, proving it is a Dedekind domain and analyzing idempotent matrix products over such rings.

Findings

D_K can be a valuation or Bézout domain.

The minimal Dress ring D of alculus(X) is Dedekind.

Products of 2x2 idempotent matrices over D are studied.

Abstract

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.