Given the ideal $I=(x^3,xy,xz^2)\subseteq k[x,y,z]$ how do I find the associated, minimal and embedded prime ideals?
I got the minimal primary decomposition to be $I=(x,y)\cap (x^3,z^2)$ so that the associated primes are $(x,y)$ and $(x,z)$, by the first uniqueness theorem.
I'm not sure how I go about finding the rest. I know that the minimal prime ideals have to be the minimal of the associated prime ideals while the rest are embedded, but which is the minimal?
Just wanted to throw it out there that Macaulay2 gives a great way to find these! And there's a great online interface here. Here's the code:
You can see the language is very intuitive!
Also notice that decompose and minimalPrimes do the same thing. Hope this helps!