I mean, I think that you should know the proposition

This follows immediately from the lattice isomorphism theorem (fourth) because you know that the ideals of sitting above are in bijective correspondence with the ideals of , and as you have noted, there aren't a lot of those.Theorem:Let be a commutative ring, and an ideal of . Then, is maximal if and only if is a field

Now, apply this theorem to your problem, recalling another one of the isomorphism theorems.