There is a statement which says, if $I$ is an ideal of the localized ring $S^{-1}A$ then there exists an ideal $J$ of the ring $A$ such that $I=JS^{-1}A$. I'm failing to prove it.
When $I$ is an ideal of $S^{-1}A$ , it means for every $i\in I$ and $\frac{a}{s} \in S^{-1}A$ : $i*\frac{a}{s}\in I$
Otherwise when $J$ is an ideal of $A$, it means for every $j \in J$ and $a \in A$: $j*a\in J$. How can I connect both ideals?
Let $\varphi:A\longrightarrow S^{-1}A$ be the canonical map $a\longmapsto \frac a1$, and consider $$J=\varphi^{-1}(I)$$ We of course have $\;\varphi(J)\subset I$, hence $\;\varphi(J)S^{-1}A\subset I$.
Conversely, if $\frac as\in I$, we also have, since $I$ is an ideal of $S^{-1}A$, $\frac a1\in I$, so by definition, $a\in J$, and $$\frac as=\frac 1s\cdot \frac a1\in S^{-1}A\cdot\varphi(J).$$