I am asked to show that if $M$ is an essential extension of $N$, then $S^{-1}M$ is an essential extension of $S^{-1}N$.
So this is how I approached the problem:
Let $U\neq (0)$ be a submodule of $S^{-1}M$. Then we can write $U = S^{-1}V$ for some submodule $V$ of $M$. Since $U\neq (0)$, $V\neq (0)$. Since $M$ is an essential extension of $N$, and $V$ is non-zero submodule of $M$, $V\cap N\neq (0)$. So choose $x\neq 0$ in $V\cap N$.
My problem is now trying to get a non-zero element of $U\cap S^{-1}N$. I initially thought I could just use $x/1$ but I do not know that this is non-zero. So now I am stuck.
Any suggestions?