about correspondence between ideal of a ring and ideal of its localized ring

Question

Let $A$ be a ring, $S$ its multiplicative subset. Let $J(A)$ denote the set of ideals of $A$.
Referring to localization of rings in Lang's Algebra pg.110, $f:J(A)\rightarrow J(S^{-1}A)$ associates ideals in $A$ with ideals in the localized ring by $f(\mathfrak a)=S^{-1}\mathfrak a$ where $\mathfrak a$ is ideal in A.

My question is whether this association is surjective. I think that for every ideal $\mathfrak a$ in $S^{-1}A$, $\mathfrak a\cap A$ is ideal in A and $f(\mathfrak a\cap A)= \mathfrak a$.

I got my own proof for this but I got suspicious as I expected that the association wouldn't be surjective. Things seemed to turn out too easy and I wanted to check if it was really so.

Answer 1

It is a bit confusing when you first denote an ideal in $A$ by $\mathfrak{a}$ and then later you use the same notation to denote an ideal in $S^{-1}A$. But assuming that you want to denote $\mathfrak{a}$ to be an ideal in $A$, then every ideal $\mathfrak{b}$ of $S^{-1}A$ is of the form $S^{-1}\mathfrak{a}$ for some ideal $\mathfrak{a}$ of $A$ so the association $\mathfrak{a}\mapsto S^{-1}\mathfrak{a}$ is indeed surjective.