Corollary 1

— Every finitely generated module E over a principal ideal domain is the direct sum of the torsion submodule of E and a free module.

The torsion submodule of E has in general several distinct complements. For example, if $E = \mathbf{Z} \times (\mathbf{Z}/(2))$ then the torsion submodule of E is $\{0\} \times (\mathbf{Z}/(2))$; it has as a complement the submodule $\mathbf{Z} \times (0)$, and also the submodule consisting of all elements $(n, \overline{n})$, where $n$ runs through $\mathbf{Z}$ and $\overline{n}$ is the residue class of $n$ mod 2.