Lemma 10.82.14. Let $R \to S$ be a ring map which is universally injective as a map of $R$-modules. Then the functor $M \mapsto M \otimes _ R S$ on $R$-modules reflects injections, surjections, and isomorphisms.
Proof. Let $M \to N$ be a map of $R$-modules with kernel $K$ and cokernel $Q$. If $M \otimes _ R S \to N \otimes _ R S$ is injective, then the image of $K \otimes _ R S \to M \otimes _ R S$ is zero. Since $K \subset K \otimes _ R S$ and $M \subset M \otimes _ R S$ we conclude that $K = 0$. On the other hand, if $M \otimes _ R S \to N \otimes _ R S$ is surjective, then $Q \otimes _ R S$ is zero (by right exactness of tensor product). Hence $Q \subset Q \otimes _ R S$ is zero too. $\square$
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