Let $M$ be an $R$-module and $N$ be the torsion sub-module. Then show that $M/N$ is a torsion-free $R$-module.
Answer:
Let $a \in M/N$,then $a=b+N, \ b \in M$.
Let $r \in R$, then $ra=r(b+N)=rb+rN=rb$, because $rN=0$ as $N$ torsion submodule.
Thus,
$ra=rb \neq N$ as $rb \neq 0$.
Thus $ra \neq N$ which implies $M/N$ is torsion-free.
But i think something is wrong.