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Here is a version of the Correspondence Theorem:

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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?

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    $\begingroup$ I am convinced. $\endgroup$ – Jesko Hüttenhain Feb 12 '18 at 1:08
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    $\begingroup$ Yes, that is correct. $\endgroup$ – B. W. Feb 12 '18 at 9:14

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