Corollary

— A submodule $M$ of finite rank in a free module $L$ over a principal ideal domain $A$ has a complement in $L$ if and only if $L/M$ is torsion free.

In the notation of Th. 1, if $L/M$ is torsion free, then $M = M'$, and $M'$ has a complement $L'$ in $L$. Conversely if $M$ has a complement $L'$ in $L$, then $L/M$ is isomorphic to $L'$, which is free (VII, p. 15, Cor. 2), and a fortiori torsion free.