Lemma 10.27.9. Let $R$ be a ring. Let $S$ be a multiplicative subset of $R$. An ideal $I \subset R$ which is maximal with respect to the property that $I \cap S = \emptyset$ is prime.

Proof. This is the example discussed in the introduction to this section. For an alternative proof, combine Example 10.27.3 with Proposition 10.27.8. $\square$

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