Let's define $M$ a left $K$-module and $M'$ a submodule of $M$. We suppose $M'$ and $M/M'$ are torsion modules. I want to show that if $K$ is an integral domain then $M$ is a torsion module.
According to the definition if $M'$ is a torsion, then for each $x \in M'$ there exists a $k \in K$ such that $km=0$.
We also know that there is a natural map $\pi:M \rightarrow M/M'$ defined as $\pi(m)=m+M'$, which satisfied $\pi(m+t)=\pi(m)+\pi(t)$, and $\pi(\lambda m) = \lambda \pi(m)$.
But i'm not sure how to combine this to show that for each $m\in M$ there exists an element $r$ in $K$ such that $rm=0$. Can anyone provide a hint or an answer?