Are $K[X,Y,Z]$, $K[X,Y,Z]/(Z^2)$ and $K[X,Y,Z]/(XZ,YZ,Z^2)$ flat $K[X,Y]$-modules?

Question

Let $K$ be a field, $A = K[X,Y]$ be the polynomial ring in two variables. Are $K[X,Y,Z]$, $K[X,Y,Z]/(Z^2)$ and $K[X,Y,Z]/(XZ,YZ,Z^2)$ flat $A$-modules?

I have no idea to check which module is flat and which is not. Can anyone give me some hints and some ideas for the proof? Many thanks

Answer 1

$K[X,Y,Z]=A[Z]$ is free over $A$, hence flat.

$A[Z]/(Z^2)$ is also free over $A$.

Since flat modules are torsion-free, the last module is not flat.