A submodule of a free module is torsion-free?

A submodule of a free module is torsion-free?

I am studying for a comprehensive exam and looking at a large bank of
problems. One problem has six statements about module and asks for a a
proof or a counter-example of the statements. I am able to solve all
except two related to torsion . Assume that R is an integral domain and
the modules below are R-modules. So an element $m \in M$ is torsion if
there is an $r \in R-0$ s.t. $rm=0$.
A submodule of a free module is torsion-free.
A submodule of a torsion module is a torsion module.
For #1, I know that a submodule of a free module is not necessarily free
and I know that a free module is torsion free but I can't put these to use
to find a counterexample. For #2, this seems logical but again I am unable
to provide a proof. By the way, to be clear a torsion module has only
torsion elements. Thanks!