Let A = K[x,y,z] and let p be the ideal ($xz - y^2, yz - x^3, x^2y - z^2$). Is $p^2$ a $p$-primary ideal?
I'm still trying to get used to the idea of p-primary modules and I am having some trouble with this example.
From what I understand, p$^2$ will be p-primary if and only if for every $a \in$ p, the map $A/{p^2} \rightarrow A/{p^2}$ given by multiplication by $a$ is injective if $a \notin p$ and nilpotent if $a \in p$.
Here the ring $A/p^2 = $ $\frac{K[x,y,z]}{((xz - y^2)^2,(yz - x^3)^2,(x^2y - z^2)^2, (xz - y^2)(yz - x^3),(xz - y^2)(x^2y - z^2),(yz - x^3) )(x^2y - z^2))}$
It may or may not be reasonable to make this simpler since basically two of the three polynomials must be zero and the sqaure of the third must be zero. But regardless, I don't see why this map should be injective / nilpotent at for the appropriate multiplication. (Or if p$^2$ is not p-primary, then I don't see any counterexample or why this should be the case).
Aside: This is an unrelated question, but I was trying to show that p is a prime ideal and I believe that it is the kenel of the m