Let $R$ be a commutative ring (with unit). Then if $$0\longrightarrow M'\longrightarrow M\longrightarrow M''\longrightarrow 0$$ is an exact sequence of $R$-modules, with $M''$ $R$-flat, $M$ is flat if and only if $M'$ is flat.
Now suppose that $M'=M''$. The preceding result gives that $M$ is flat if $M'=M''$ is. My question is: if $M$ is flat, does this implies that $M'$ is flat ?