Converse to Exactness of localization of modules

Question

It is a standard fact that if $R$ is a ring, and $$\tag{1} 0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0 $$ a short exact sequence of $R$-modules, if $S$ is a multiplicative subset of $R$, then passing to quotients, the sequence $$\tag{2} 0\longrightarrow S^{-1}A\overset{S^{-1}f}{\longrightarrow}S^{-1}B\overset{S^{-1}g}{\longrightarrow}S^{-1}C\longrightarrow 0 $$ is exact as well. However, I cannot find a counterexample for the converse, that is, non-exact sequences (1) such that (2) is a short exact sequence. Any insight greatly appreciated!

Answer 1

What you're asking is if $S^{-1}R$ is a faithfully flat $R$-module. It isn't, take $R=\mathbb{Z}$, $A=B=C=\mathbb{Z}/2\mathbb{Z}$, and the maps $A\to B$ and $B\to C$ being the zero maps, and consider $S=\mathbb{Z}-\{0\}$.