Example 9

Suppose that $A = K[X, Y]$, where $K$ is a field. Then the maximal ideal $m$ generated by $X$ and $Y$ is a torsion-free $A$-module, but not flat. Consider indeed the ring $B = A/(Y)$, which is isomorphic to $K[X]$, hence integral. The $B$-module $m_{(B)}$ is isomorphic to $m/Ym = (X, Y)/(XY, Y^2)$ in which the class of $Y$ is torsion. Consequently, $m_{(B)}$ is not a flat $B$-module, hence $m$ is not flat.