I am studying localization of rings and got stuck at a problem. It states that if $S$ is a multiplicatively closed subset of a ring $A$ then fractional ideals of $\ S^{-1}A $ are in bijective correspondence with those of $A$ which do not meet $S$. However, prime ideals of $ S^{-1}A $ are in bijective correspondence with those of $A$ which do not meet $S$.
My question is why can't we say that ideals of $A$ which do not meet $S$ are in bijective correspondence with those of $S^{-1}A$?
We know that ideals of $S ^{-1}A$ are of the form $S^{-1}I$ where $I$ is an ideal of $A$. Why won't the correspondence $I\rightarrow S^{-1}I $ work?
The correspondence $I\mapsto S^{-1}I$ need not be injective. For instance, let $A=k[x,y]$ and $S=\{y^n:y\in\mathbb{N}\}$. Then if $I=(x)$ and $J=(xy)$, $S^{-1}I= S^{-1}J$ even though $I\neq J$ and neither intersects $S$.