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A simple proof of uniqueness for torsion modules over principal ideal domains.

The aim of this note is to give an alternative proof of uniqueness for the decomposition of a finitely generated torsion module over a P.I.D. (= principal ideal domain) as a direct sum of indecomposable submodules.
Our proof tries to mimic as far as we can the standard procedures used when dealing with vector spaces.
For the sake of completeness we also include a proof of the existence theorem.