Prove that if R is an integral domain then Torsion of M is a submodule of R-module M I can't quite get this one.  I can show that two non-zero elements m,n of M have a non-zero product but they belong to the R-module, not the integral domain so I don't know if it's necessarily true that their product be nonzero.

Let $M$ be an left $R$-module such that $r\cdot m=rm$ is well defined. Then if $m_{1},m_{2}$ are torsion elements, $r_{1}m_{1}+r_{2}m_{2}$ must also be a torsion element for all $r_{1},r_{2}\in R$. We need to show that $M_{1}$ generated by $m$ of the kind
$$
m=\sum^{n}_{i=1}r_{i}m_{i}, \exists R_{i}m_{i}=0,r_{i},R_{i}\in R
$$
is a well defined submodule. But this just follows from definition. Note that we used "integral domain" implicitly as $R_{i}\in R$ are not zero divisors. 
