Let $\varphi: R \to S$ be a homomorphism of commutative rings, and let $M$ be a finitely generated $R$-module. Regard $S$ as an $R$-module via $\varphi$, and equip $S \otimes_R M$ with its standard $S$-module structure
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\begin{align*}
a \cdot (s \otimes m) = (as) \otimes m
\end{align*}
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for $a,s \in S$ and $m \in M$. Then $S \otimes_R M$ is finitely generated as an $S$-module.