[Math] Showing that if $I$ is a maximal ideal, $R/I$ is a Field

Question

From a Masters Qual. Practice Exam:

Let $R$ be a commutative ring with identity, and let $I$ be an ideal
of $R$. Prove there is a bijection between the intermediate ideals $J$
such that $I \subseteq J \subseteq R$ and the ideals of the quotient
ring $R/I$. Thus prove that if $I$ is maximal ideal, then $R/I$ is a
field.

I've read other proofs that if $I$ is a maximal ideal, $R/I$ is a Field, but I'm having a hard time understanding them, I can't even tell if they use this same technique or not.

Edit: I think now I have the bijection, we let $\phi(I) = J/I$, and this gives us a correspondence between intermediate ideals and ideals of $R/I$.

I still don't know how to get that $R/I$ is a field after this.

Answer 1

If $I$ is amaximal ideal, then there are only two intermediate ideals, namely, $I$ and $R$. Thus, $R/I$ has only two ideals.

What kind of rings have only two ideals?