Let $A$ be a ring and $S$ a Multiplicatively closed set. Let us consider a map $f$ : $A \rightarrow S^{-1} A$ by $a \rightarrow \frac{a}{1}$.
Now for an ideal $I$ of $A$, the extension of $I$ denoted by $I^e$ is the set of all linear combinations of $\frac{a}{1}$ in $S^-1 A$ which eventually turns out to be elements of form $\frac{x}{s}$ where $x \in I$.
Contraction on an ideal $S^{-1} J$ in {$S^{-1} A$ is ${j : \frac{j}{1} \in S^{-1} J}$} and is denoted by $J^c$.
Now for any prime ideal P in $S^{-1} A$, $P^{ce}$ is P itself.
Is it true for any ideal I or only prime ideals?
I not, what would a suitable counter-example be?