Lemma 15.24.2. Let $A$ be a ring. Let $M$ be a flat $A$-module. Let $x \in M$. The content ideal of $x$, if it exists, is finitely generated.

**Proof.**
Say $x \in IM$. Then we can write $x = \sum _{i = 1, \ldots , n} f_ i x_ i$ with $f_ i \in I$ and $x_ i \in M$. Hence $x \in I'M$ with $I' = (f_1, \ldots , f_ n)$.
$\square$

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