Here is a version of the Correspondence Theorem:
I need to generalize it for prime and maximal ideals. That is, under the bijection mentioned in the statement of the theorem, I need to show that prime ideals correspond to prime ideals and maximal ideals correspond to maximal ideals.
This seems obvious so I have some doubts. I would prove it like this. If $I\subset R$ is prime (maximal) then $R/I$ is a domain (field). Let $\mathcal I=\varphi(I)$. Since $R/I\simeq \mathcal R/\mathcal I$ and $R/I$ is a domain (field), $\mathcal R/\mathcal I$ is also a domain (field). Hence $\mathcal I$ is prime (maximal) in $\mathcal R$. The converse is similar.
Is that correct?