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!