I have some problem in understanding the following. Let $M$ be an $A$ module and $S \subset A$ be any multiplicative closed set. Now consider the canonical $A$ linear map $\phi : M \to S^{-1}M,$ i.e., $\phi(m)=\frac{m}{1}$ and let $N$ be any submodule of $M.$ Then clearly $\phi(N)$ is an $A$ submodule of $S^{-1}M.$ Then what do we mean by $S^{-1}N ?$ Is it the $S^{-1}A$ submodule of $S^{-1}M$ generated by $\phi(N)$ ? Also can we say that $\phi^{-1}(S^{-1}N)=N $ ?
Since $\phi(N) \subset S^{-1}N ,$ $N \subset \phi^{-1}(\phi(N))\subset \phi^{-1}(S^{-1}N).$ Conversely if $x \in \phi^{-1}(S^{-1}N) $ then $\phi(x) \in S^{-1}N,$ i.e., $\frac{x}{1}= \frac{y}{s}$ for $y \in N$ and $s \in S.$ Then $\exists \;t \in S$ such that $stx=ty,$ which imply that $stx \in N.$ After this I cannot go further. It is understandable that if $N$ is $p$ primary and $p \cap S =\phi$ then we are done.