On coprime rational function matrices

The concept of coprimeness of matrices with elements in a field of fractions is introduced. We focus on the field of rational functions and define when two rational matrices are coprime with respect to different rings. The definition of coprimeness at infinity of polynomial matrices is obtained as a particular case. Moreover, the fact that coprimeness is a local property is proved and it is shown how coprimeness changes by performing Möbius transformations.