Example 10.27.4. Let $R$ be a ring, $I \subset R$ an ideal, and $a \in R$. If $(I : a)$ is generated by $a_1, \ldots , a_ n$ and $(I, a)$ is generated by $a, b_1, \ldots , b_ m$ with $b_1, \ldots , b_ m \in I$, then $I$ is generated by $aa_1, \ldots , aa_ n, b_1, \ldots , b_ m$. To see this, note that if $x \in I$, then $x \in (I, a)$ is a linear combination of $a, b_1, \ldots , b_ m$, but the coefficient of $a$ must lie in $(I:a)$. As a result, we deduce that the family of finitely generated ideals is an Oka family.

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