The Kaplansky condition and rings of almost stable range 1

TL;DR

This paper explores conditions under which K-Hermite rings are elementary divisor rings, introduces variants of the Kaplansky condition, and provides a counterexample to a previous conjecture about stable range properties.

Contribution

It introduces new variants of the Kaplansky condition for K-Hermite rings and presents a counterexample to a conjecture relating elementary divisor rings and stable range 1.

Findings

A commutative K-Hermite ring is an EDR iff a specific element condition holds.

An example of a Bézout domain is an EDR but lacks almost stable range 1.

The paper answers a question of Warren Wm. McGovern with this example.

Abstract

We present some variants of the Kaplansky condition for a K-Hermite ring $R$ to be an elementary divisor ring; for example, a commutative K-Hermite ring $R$ is an EDR iff for any elements $x,y,z\in R$ such that $(x,y)=(1)$, there exists an element $\lambda\in R$ such that $x+\lambda y=uv$, where $(u,z)=(v,1-z)=(1)$. We present an example of a a B\'ezout domain that is an elementary divisor ring, but it does not have almost stable range 1, thus answering a question of Warren Wm. McGovern.