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 {\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 Bézout 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 Prüfer non-Bézout domains.

References in corpus (1)

• Factorizations into idempotent factors of matrices over Prüfer domains

Cited by in corpus (1)

• PRINC domains and comaximal factorization domains