Conditions for stable range of an elementary divisor rings

TL;DR

This paper establishes that a commutative Bezout domain is an elementary divisor ring if and only if it has Gelfand range 1, providing a unified criterion for various classes of domains.

Contribution

It proves a necessary and sufficient condition for commutative Bezout domains to be elementary divisor rings using Gelfand range 1, extending known results to specific classes.

Findings

Characterization of elementary divisor rings via Gelfand range 1

Solution to the problem for classes like $PM^{*}$ and local Gelfand domains

Unified criterion for commutative Bezout domains

Abstract

Using the concept of ring of Gelfand range 1 we proved that a commutative Bezout domain is an elementary divisor ring iff it is a ring of Gelfand range 1. Obtained results give a solution of problem of elementary divisor rings for different classes of commutative Bezout domains, in particular, $PM^{*}$, local Gelfand domains and so on.