Lemma 10.28.10. Let $R$ be a ring.

1. An ideal $I \subset R$ maximal with respect to not being finitely generated is prime.

2. If every prime ideal of $R$ is finitely generated, then every ideal of $R$ is finitely generated1.

Proof. The first assertion is an immediate consequence of Example 10.28.4 and Proposition 10.28.8. For the second, suppose that there exists an ideal $I \subset R$ which is not finitely generated. The union of a totally ordered chain $\left\{ I_\alpha \right\}$ of ideals that are not finitely generated is not finitely generated; indeed, if $I = \bigcup I_\alpha$ were generated by $a_1, \ldots , a_ n$, then all the generators would belong to some $I_\alpha$ and would consequently generate it. By Zorn's lemma, there is an ideal maximal with respect to being not finitely generated. By the first part this ideal is prime. $\square$

[1] Later we will say that $R$ is Noetherian.

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