Primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field

Question

I am looking for the primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field.

I am not looking for a solution here, rather a hint or two. Is there a general strategy for approaching primary decomposition? I realize that you have to make an educated guess first, but I am not sure where to start. Possibly a resource with a few worked out examples would help.

Answer 1

Yes, there is a strategy given by the fact that your ideal is monomial.

So, since $\gcd(x,z)=1$ we have $(x^2,xy,xz)=(x^2,xy,x)\cap(x^2,xy,z)$. The first ideal is $(x)$ and thus we get $(x^2,xy,xz)=(x)\cap (x^2,xy,z)$. Now apply the same trick to $(x^2,xy,z)$, and so on.

For another example you can look here.