The Stacks project

Lemma 15.23.5. Let $R \subset R'$ be an extension of domains. If $M$ is an $R$-module, then the rank of $M$ over $R$ is equal to the rank of $M \otimes _ R R'$ over $R'$.

Proof. This is true because the dimension of a vector space is invariant under extension of ground field. $\square$

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