Bezout domains with stable range 1

Abstract

It is shown that certain classes of Bezout domains have stable range 1, and thus are elementary divisor rings. Included is a strengthening of Roquette's principal ideal theorem which states that the holomorphy ring of a family S of valuation rings of a field K, with S having bounded residue fields, is Bezout. A counterpart is also given where a bound is placed on the ramification indices instead of the residue fields, and these results are applied to rings of integer-valued rational functions over these rings. Along the way, characterizations are given of Prüfer domains with torsion class group, Bezout domains, and Bezout domains with stable range 1 in terms of a family { B(t) | t∈K} of numerical semigroups associated with the ring R, and a related family { D(t) | t∈K} of numerical semigroups.