Elementary matrix reduction over locally stable rings

TL;DR

This paper characterizes elementary divisor rings among Bezout rings by introducing the concept of locally stable rings, linking stable range conditions to matrix reduction properties.

Contribution

It establishes that for Bezout rings, being locally stable, having neat range 1, and being an elementary divisor ring are equivalent conditions.

Findings

R is an elementary divisor ring iff it is locally stable.

Locally stable rings have stable range 1.

Equivalence of properties in Bezout rings.

Abstract

A commutative ring R is locally stable provided that for any $a,b\in R$ such that $aR+bR=R$, there exist some $y\in R$ such that $R/(a+by)R$ has stable range 1.For a Bezout ring $R$, we prove that $R$ is an elementary divisor ring if and only if $R$ is locally stable if and only if $R$ has neat range 1.