TITLE: Correspondence Theorem for Rings
QUESTION [5 upvotes]: Let $I$ a proper ideal of a ring $R$. For each intermediate ideal $I\subseteq J\subseteq R$, we consider the map $$J\mapsto \pi(J):=J/I:=\{a+I:a\in J\}.$$
I want to prove that this is a bijection from the intermediate ideals to the set of ideals of $R/I$.
It is easy to see that $J/I$ is an ideal of $R/I$.
For injectivity, if $J/I=J'/I$ and $a\in J$, then $a+I=a'+I$ for some $a'\in J'$, so $a'-a\in I\subseteq J'$. It follows $a\in J'$. Similarly $J'\subseteq J$.
What I can't do yet is to prove this is a surjective map.
Let $I_0$ an ideal of $R/I$. How can I find an ideal $J$, with $I\subseteq J\subseteq R$ such that $J/I=I_0$?
Thank you.

REPLY [3 votes]: Take the preimage of $I_0$ under the quotient map $R\to R/I$.