Lemma 30.9.3. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Any quasi-coherent submodule of $\mathcal{F}$ is coherent. Any quasi-coherent quotient module of $\mathcal{F}$ is coherent.

**Proof.**
We may assume that $X$ is affine, say $X = \mathop{\mathrm{Spec}}(A)$. Properties, Lemma 28.5.2 implies that $A$ is Noetherian. Lemma 30.9.1 turns this into algebra. The algebraic counter part of the lemma is that a quotient, or a submodule of a finite $A$-module is a finite $A$-module, see for example Algebra, Lemma 10.51.1.
$\square$

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