I am wondering how to solve this problem: given $f:A\rightarrow B$ and $g:B\rightarrow C$ ring homomorphisms. If $g\circ f$ is flat, and $g$ is faithfully flat, then $f$ is flat.
If I am not mistaken, the question asks us to prove that $B$ is a flat $A$-module. So we want to show that if $M$ and $N$ are $A$ modules and if $M\rightarrow N$ is injective, then $M\otimes_{A}B\rightarrow N\otimes_{A}B$ is also injective.
So I proceed as follows: since $C$ is flat over $A$, so $M\otimes_{A}C\rightarrow N\otimes_{A}C$ is also injective. But I am not sure from here how to use the fact that $C$ is faithfully flat over $B$. Here are some approaches that I tried:
1) $C\cong C\otimes_{B}B$, BUT to use associativity property on $M\otimes_{A}(C\otimes_{B}B)$, I require $C$ and $B$ to be $A$ modules.
2) So I attempted to see if $C\cong C\otimes_{A}B$ as $A$ modules, but I couldn't.
Any other approaches?