TITLE: A submodule of a free module is torsion-free?
QUESTION [7 upvotes]: 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!

REPLY [4 votes]: Number 1 is true.
Consider $r \in R-0$ and $t \in S \subset M$.  Thus, since $M$ is free. $t=r_1m_1+ \ldots + r_nm_n$.  Assume $rt=0$.  Then $rr_1m_1+ \ldots + rr_nm_n$, then $rr_i=0$ and since $R$ is an integral domain, then $r_i=0$ implying $t=0$ and $S$ is torsion-free.  QED
Thanks to @danielfischer for a good hint.
Number 2 is true too.
Consider $t \in S \subset M$, since $t\in M$, $\exists r \in R-0$ s.t. $rt=0$.  Thus, S is a torsion module. QED
I'm not sure how I missed this one to begin with, had a brain freeze I guess.