The Stacks project

Proof. Suppose $M$ is finite. Then choose a presentation $0 \to K \to R^{\oplus n} \to M \to 0$. This gives an exact sequence $K \otimes _ R N \to N^{\oplus n} \to M \otimes _ R N \to 0$ by Lemma 10.12.10. We conclude that if $N$ is finite too then $M \otimes _ R N$ is a quotient of a finite module, hence finite, see Lemma 10.5.3. Similarly, if both $N$ and $M$ are finitely presented, then we see that $K$ is finite and that $M \otimes _ R N$ is a quotient of the finitely presented module $N^{\oplus n}$ by a finite module, namely $K \otimes _ R N$, and hence finitely presented, see Lemma 10.5.3. $\square$