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Here is a version of the Correspondence Theorem:

enter image description here

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, 2018 at 1:08
  • 1
    $\begingroup$ Yes, that is correct. $\endgroup$
    – aaa
    Feb 12, 2018 at 9:14

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