 Solution

Localization of the inclusion $N\hookrightarrow M$ gives an injection $S^{-1}N\hookrightarrow S^{-1}M$, whose image lies in $N^{\prime }$. Conversely, take $m/s\in N^{\prime }$. Since $s/1$ is a unit and $N^{\prime }$ is an $S^{-1}A$-submodule,By the definition of the inverse image, $m\in N$, and hence $m/s$ lies in the image of $S^{-1}N$. Therefore