To Show That a Certain Module is Not Flat

Question

Let $k$ be a field and $R=k[x]$. Let $M=\dfrac{k[x, y]}{(x)\cap(x, y)^2}$. How to show that $M$ is not a flat $R$-module?

I am sorry but I am really blank on this one.

Answer 1

If $M$ is flat, then it must be torsion-free (see here), but this is not the case: $xy=0$ and $y\ne 0$ in $M$.

Answer 2

Let $M=k[x]/(x^2)$, $N=k[x]/(x)$ and $f:M\to N$ the map induced by the identity of $k[x]$. Let $K$ be the kernel of $f$, so that we have an exact sequence $$0\to K\to M\to N\to 0$$ What happens if you tensor this with $k[x,y]/(x^2,xy)$?