The Stacks project

Example 10.28.3. Let $R$ be a ring and let $S$ be a multiplicative subset of $R$. We claim that $\mathcal{F} = \{ I \subset R \mid I \cap S \not= \emptyset \} $ is an Oka family. Namely, suppose that $(I : a), (I, a) \in \mathcal{F}$ for some $a \in R$. Then pick $s \in (I, a) \cap S$ and $s' \in (I : a) \cap S$. Then $ss' \in I \cap S$ and hence $I \in \mathcal{F}$. Thus $\mathcal{F}$ is an Oka family.