One sided invertibility of matrices over commutative rings, corona problems, and Toeplitz operators with matrix symbols

Conditions are established under which Fredholmness, Coburn's property and one- or two-sided invertibility are shared by a Toeplitz operator with matrix symbol G   and the Toeplitz operator with scalar symbol det⁡Gdet⁡G. These results are based on one-sided invertibility criteria for rectangular matrices over appropriate commutative rings and related scalar corona type problems.