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!
Best Answer
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.