Lemma 15.23.4. Let $R$ be a domain. Let $M$ and $N$ be $R$-modules.
The rank of $M \otimes _ R N$ is the product of the ranks of $M$ and $N$.
If $M$ is a finitely presented $R$-module, then the rank of $\mathop{\mathrm{Hom}}\nolimits _ R(M, N)$ is the product of the ranks of $M$ and $N$.
Proof. Part (1) follows from the same fact for vector spaces. Assume $M$ is finitely presented as an $R$-module. Then $\mathop{\mathrm{Hom}}\nolimits _ R(M, N) \otimes _ R K$ is isomorphic to $\mathop{\mathrm{Hom}}\nolimits _ K(M \otimes _ R K, N \otimes _ R K)$ as a $K$-vector space by Algebra, Lemma 10.10.2. Since also $M \otimes _ R K$ is finite dimensional, we conclude from the corresponding fact for vector spaces. $\square$
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