# Rings of the right (left) almost stable range 1

## TL;DR

This paper introduces the concept of rings with right or left almost stable range 1, develops a canonical matrix reduction theory for such rings, and characterizes new classes of noncommutative elementary divisor rings.

## Contribution

It defines rings of almost stable range 1, constructs a canonical diagonal reduction for matrices over these rings, and characterizes when certain classes of rings are elementary divisor rings.

## Key findings

- Every D-adequate Bézout domain has almost stable range 1.
- Necessary and sufficient conditions for Hermite D-rings to be elementary divisor rings.
- Every L-ring of almost stable range 1 is a ring of right almost stable range 1.

## Abstract

We introduce a concept of rings of right (left) almost stable range $1$ and we construct a theory of a canonical diagonal reduction of matrices over such rings. A description of new classes of noncommutative elementary divisor rings is done as well. In particular, for B\'ezout $D$-domain we introduced the notions of $D$-adequate element and $D$-adequate ring. We proved that every $D$-adequate B\'ezout domain has almost stable range $1$. For Hermite $D$-ring we proved the necessary and sufficient conditions to be an elementary divisor ring. A ring $R$ is called an $L$-ring if the condition $RaR = R$ for some $a\in R$ implies that $a$ is a unit of $R$. We proved that every $L$-ring of almost stable range $1$ is a ring of right almost stable range $1$.