Ideal $(Y^2,X-YZ)$ is $(X,Y)$-primary

Question

Show that the ideal $(Y^2,X-YZ)$ is $(X,Y)$-primary in $K[X,Y,Z]$, where $K$ is a field.

I got a hint that I need to use this property: Let $f:A\to B$ be a ring homomorphism. If $q$ is $p$-primary in $B$ then $f^{-1}(q)$ is $f^{-1}(q)$-primary in $A$. But I am unable to choose a map $f$ and a ring $B$. Can anyone give me a hint? Thank you.

Answer 1

I'd try $A=K[X,Y,Z]$ and $B=K[X,Y,Z]/(X-YZ)$. Note that $B\simeq K[Y,Z]$ and the image of $(Y^2,X-YZ)$ in $K[Y,Z]$ is $(Y^2)$ which is $(Y)$-primary.