The structure of the Smith-MacMillan form of a rational matrixT(s) inside a regionP=ω∪{∞} (where ω is asymmetric with respect to the real axis subset of the finite complex plane C) is determined. Algorithmic procedures based on elementary row and column operations over the euclidean ringRP(s) consisting of all rational functions with no poles inPare given. Coprimeness inPof a pair of rational matrices is studied in detail. These results lead to constructive procedures for determining the coprime inPmatrix fraction descriptions ofT(a).