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Here I will quote a few basic examples of maximality condition yielding primeness of ideals, though I knew how to prove them and knew the statement. I do not think I have a good intuitive answer to expect primeness from maximality condition in general. In general, the proof goes by contradiction and produce an ideal strictly larger by assuming contradiction. Then deduce the "maximal" ideal does not have the desired property. The proof will almost all the time invoke zorn lemma.

Every maximal ideal is prime.

A maximal among non finitely generated ideals is prime.

Maximal among non principal is prime.

Let $M$ be a noetherian module over ring $A$. Denote $Q$ as an ideal maximal among the ideals annhilates some principal submodule of $M$. Then $Q$ is prime.

$\textbf{Q:}$ The proof above is not too hard. Was there intuitive explanation or reason for why I should expect maximal implying prime in general? I am not looking for a proof explanation.

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    $\begingroup$ See the links I give here for a start. See also here. $\endgroup$ – Bill Dubuque Apr 10 at 15:18
  • $\begingroup$ @BillDubuque Thanks a lot for the link. $\endgroup$ – user45765 Apr 10 at 15:20
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    $\begingroup$ The Lam-Reyes paper on Oka-Ako families is really "the rug that ties the room together" for primeness results like this. $\endgroup$ – rschwieb Apr 10 at 15:32

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