Take the ring $R=\mathbb{C}[x,y,z]/(x^3-y^3-z^3)$. I want to find a prime ideal, $p$, where $p^3$ is not a primary ideal. Any suggestions how to find such a prime ideal?
I tried a few, like $(x,z)$, $(xy-z)$, $(y^2)$, and $(xyz)$ (and more) but can't find anything. I think I need to find a prime ideal such that it's third power will eliminate one of the cubes in the quotient, but not certain.
The wiki on prime ideals give the example of $R=\mathbb{C}[x,y,z]/(xy-z^2)$, but I've found this is a vveerryy different ring than the one I am looking at.
Any advice or hints welcome. Thank you!