Elementary divisor rings with Dubrovin-Komarnytsky property

TL;DR

This paper introduces noncommutative rings with the Dubrovin-Komarnytsky property, explores their elementary divisor rings with stable range 1, and constructs new families of such rings through reduction matrices.

Contribution

It defines noncommutative rings with DK-property, develops a theory of reduction matrices, and constructs new examples of elementary divisor rings with stable range 1.

Findings

New families of noncommutative elementary divisor rings

Development of reduction matrix theory over these rings

Identification of rings with stable range 1

Abstract

We introduce noncommutative rings with $DK$-property (Dubrovin-Komarnytsky's property) and investigate elementary divisor rings with such property. Mostly we pay attention to these kinds of noncommutative rings which have stable range $1$. A theory of reduction matrices over such rings is constructed. As a consequence, new families of non-commutative rings of elementary divisor rings are constructed.