Grigore Calugareanu, Horia F. Pop: On stable range one matrices, 317-327


For $2\times 2$ 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 and we show that all matrices with one zero row (or zero column) over Bézout rings have stable range 1. Using diagonal reduction, we characterize all the $2\times 2$ integral matrices which have stable range 1 and discuss additional properties including Jacobson's Lemma for stable range 1 elements. Finally, we give an example of exchange stable range 1 integral $2\times 2$ matrix which is not clean.

Key Words: Stable range 1, clean, exchange, $2\times 2$ matrix.

2010 Mathematics Subject Classification: Primary 16U99; Secondary 16U10, 15B33, 15B36, 16-04, 15-04.

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