TITLE: Every ideal of the localization is an extended ideal
QUESTION [5 upvotes]: Let $R$ be a ring, commutative with $1$. Let $S$ be a multiplicatively closed subset of $R$, with $0\notin S,1\in R$. Let $R_S$ be the localization of $R$ at $S$. For every ideal $\mathfrak{a}\subseteq R$ of $R$, define
$$\mathfrak{a}^e:=f(\mathfrak{a})R_S$$
where $f:R\longrightarrow R_S$ is the canonical morphism sending $r$ to $\frac{r}{1}$ and $f(\mathfrak{a})R_S$ means the ideal generated by $f(\mathfrak{a})$ in $R_S$. 
I want to show (or disprove) that every ideal $\mathfrak{b}$ of $R_S$ is extended, meaning that it is of the form
$$\mathfrak{b}=\mathfrak{a}^e$$
for some ideal $\mathfrak{a}$ in $R$.

REPLY [5 votes]: Hint $\ $ Every ideal $\rm\:I\subset S^{-1}R\:$ can be generated by elements of $\rm\,R,\,$ since $\rm\: r/s\in I\iff r\in I.$