On unit stable range one matrices

TL;DR

This paper characterizes 2x2 and 3x3 matrices over commutative rings with stable range one, focusing on the Goodearl-Menal condition, and shows equivalences for integral matrices.

Contribution

It provides a complete characterization of unit stable range one matrices over commutative rings, including the special case of integral matrices.

Findings

2x2 matrices satisfying the Goodearl-Menal condition are only the zero matrix.

For integral matrices, stable range one and unit stable range one are equivalent.

The paper characterizes matrices over rings with specific stable range properties.

Abstract

We characterize the unit stable range one 2x2 and 3x3 matrices over commutative rings. In particular, we characterize the 2x2 matrices which satisfy the Goodearl-Menal condition. For 2x2 integral matrices we show that the stable range one and the unit stable range one properties are equivalent, and, that the only matrix which satisfies the Goodearl-Menal condition is the zero matrix.