How to prove that $S^{-1} A$ is a flat A-module

Solution 1:

Hint. For any $A$-module, there is a canonical isomorphism of $S^{-1}A$ modules $M\otimes_A S^{-1}A\simeq S^{-1}M$, and your problem boils down to prove that , if $0\longrightarrow J\overset{f}{\longrightarrow}K\overset{g}{\longrightarrow}L\longrightarrow 0$ is an exact sequence of $A$-modules, then

$0\longrightarrow S^{-1}J\overset{S^{-1}f}{\longrightarrow}S^{-1}K\overset{S^{-1}g}{\longrightarrow}S^{-1}L\longrightarrow 0$ is exact.

Here, we have denoted by $S^{-1}u$ the map of $S^{-1}A$-modules induced canonically by the $A$-linear map $u:M\to N$.

Now, it is enough to prove that $\ker(S^{-1}u)=S^{-1}(\ker(u)), im(S^{-1}u)=S^{-1}(im(u))$ (why?), which is not difficult...