In this paper, we give an explicit characterization of isomorphic finitely presented modules in terms of certain inflations of their presentation matrices. In particular cases, this result yields a characterization of isomorphic modules as the completion problem characterizing Serre's reduction, i.e., of the possibility to find a presentation of the module defined by fewer generators and fewer relations. This completion problem is shown to induce different isomorphisms between the modules finitely presented by the matrices defining the inflations. Finally, we show how Serre's reduction implies the existence of a certain idempotent endomorphism of the finitely presented module, i.e., that Serre's reduction implies a particular decomposition, proving the converse of a result obtained in [7].