How to prove this isomorphism?

Question

I know about the correspondence between ideals in $A$ that contains $\mathfrak a$ and ideals in the quotient ring $A/\mathfrak a$, but never see the isomorphism above, anyone knows how to prove it?

Answer 1

Write $\mathfrak{b}=\mathfrak{c}/\mathfrak{a}$ where $\mathfrak{c}$ contains $\mathfrak{a}$, then we have $A\rightarrow A/\mathfrak{a}\rightarrow (A/\mathfrak{a})/(\mathfrak{c}/\mathfrak{a})$ is surjective as it is a composite of surjective morphism. Then it is easy to see that the kernel is $\mathfrak{c}$.

In fact $\mathfrak{c}=f^{-1}(\mathfrak{b})$