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In this paper, we first present an algorithm for computing the Hermite normalform of pseudo-matrices over Pr\"ufer domains. This algorithm allows us toprovide constructive proofs of the main theoretical results on finitelypresented modules over Pr\"ufer domains and to discuss the resolution of linearsystems. In some sense, we generalize the methodology developed by Henri Cohenfor Dedekind domains. Finally, we present some results over Pr\"ufer domains ofdimension one about the Smith normal form.

Calcul matriciel g\'en\'eralis\'e sur les domaines de Pr\"ufer