On stable range one 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.