Suppose $M$ is module and $N$ a submodule over some commutative ring $R$. If $x\neq 0$, and $(x)\cap N=0$, why does this imply $(x)$ is isomorphic to some submodule of $M/N$? (Let's also assume that $M\neq 0$ and $N$ is a proper submodule.)
It was a quick detail in a passage I was reading, and I can't recall why it is.