$(X)$ faithful flat $k[X,Y]$-module

Question

I want to prove that the ideal $(X)$ is a $k[X,Y]$-faithful flat module. I succeeded to prove that it is flat but how to prove faithful flatness? Some hint?

Answer 1

The ideal $(X)$ in $k[X,Y]$ is isomorphic to $k[X,Y]$ as a $k[X,Y]$ through the map $fX\mapsto f$ (since $X$ is not a zero divisor). It follows that for any $k[X,Y]$-module $M$ we have $M\otimes_{k[X,Y]} (X)\cong M$, from which it is easy to see $(X)$ is faithfully flat.