MINIMAL PRUFER-DRESS RINGS AND PRODUCTS OF IDEMPOTENT MATRICES

摘要

We investigate a special class of Prufer domains, firstly introduced by Dress in 1965. The 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 is an element of K. We show that, for some choices of K, D-K may be a valuation domain, or, more generally, a Bezout 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 x 2 idempotent matrices over D, a subject of particular interest for Prufer non-Bezout domains.