Isomorphisms in faithfully flat $R$-modules

Question

Given that an $R$-module $M$ is faithfully flat with $R$ commutative. Let $\alpha:M'\to M''$ be any $R$-homomorphism.

I want to prove that If id${M}\otimes \alpha$ is an isomorphism, then so is $\alpha$.

Answer 1

Consider the exact sequence $0\rightarrow ker(\alpha)\rightarrow M'\rightarrow M"\rightarrow 0$. Since $M$ is flat, $0\rightarrow M\otimes ker(\alpha)\rightarrow M\otimes M'\rightarrow M\otimes M"\rightarrow 0$ is exact. Since $Id_M\otimes \alpha: M\otimes M'\rightarrow M\otimes M"$ is an isomorphism, we deduce that $M\otimes ker(\alpha)=0$. Since $M$ is faithfully flat, the fact that $M\otimes ker(\alpha)=0$ implies that $ker(\alpha)=0$.