On stable range one matrices

TL;DR

This paper characterizes 2x2 matrices with stable range 1 over various rings, explores symmetry properties, and provides examples including non-clean matrices, advancing understanding of stable range conditions in matrix rings.

Contribution

It offers a comprehensive characterization of stable range 1 matrices over commutative rings, including integral and Bezout rings, and examines symmetry and additional properties.

Findings

Characterization of 2x2 stable range 1 matrices over commutative rings

Symmetry of stable range 1 property at element level

Existence of non-clean stable range 1 matrices

Abstract

For 2 by 2 matrices over commutative rings, we prove a characterization theorem for left stable range 1 elements, we show that the stable range 1 property is left-right symmetric (also) at element level, we show that all matrices with one zero row (or zero column) over Bezout rings have stable range 1. Using diagonal reduction, we characterize all the 2 by 2 integral matrices which have stable range 1 and discuss additional properties including Jacobson Lemma for stable range 1 elements. Finally, we give an example of exchange stable range 1 integral 2 by 2 matrix which is not clean.