The Stacks project

Lemma 10.95.4. Let $R \to S$ be a ring map, let $M$ be an $R$-module, and let $Q$ be a countably generated $S$-submodule of $M \otimes _ R S$. Then there exists a countably generated $R$-submodule $P$ of $M$ such that $\mathop{\mathrm{Im}}(P \otimes _ R S \to M \otimes _ R S)$ contains $Q$.

Proof. Let $y_1, y_2, \ldots $ be generators for $Q$ and write $y_ j = \sum _ k x_{jk} \otimes s_{jk}$ for some $x_{jk} \in M$ and $s_{jk} \in S$. Then take $P$ be the submodule of $M$ generated by the $x_{jk}$. $\square$

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