Flat modules and their relationship with short exact sequences

Question

I recently came across the following result on a Wikipedia page:

Suppose $0\to A\to B\to C\to 0$ is a short exact sequence where $B,\,C$ are flat modules; then $A$ is a flat module.

I wanted to check if this works for arbitrary rings (as opposed to just commutative rings), and whether anyone could point me in the direction of a proof for this result. Many thanks in advance.

Answer 1

This follows from the long exact sequence for Tor.

If $B$ and $C$ are, say, flat left $R$-modules, and $M$ is a right $R$-module, then there is an exact sequence $$\def\Tor{\operatorname{Tor}}\Tor^R_2(M,C)\to\Tor^R_1(M,A)\to\Tor^R_1(M,B)$$ and $\Tor^R_2(M,C)$ and $\Tor^R_1(M,B)$ are zero by hypothesis. It follows that $\Tor^R_1(M,A)=0$ for all right $R$-modules $M$ so that $A$ is flat.

Answer 2

Not only this result is true, but even we have that, if $C$ is flat, $A$ is flat if and only if $B$ is flat (as right $R$-modules). You can see a proof, which requires only diagram-chasing, in Bourbaki, Commutative algebra, ch. 1 ‘Flat modules’, §4, no. 5, Prop. 5.