How to check if an ideal is primary

Question

I've the ring $A=k[x,y,z]/(z^2-xy)$ and the ideal $I=(x,y)^2$. How do I check if $I$ is prime in $A$? I know that $(x,y)$ is a prime ideal as $A/(x,y)$ is isomorphic to $k[z]$ but that does not imply $(x,y)^2$ is primary. So, how do I show that $(x,y)^2$ is primary in $A$? In fact, I think it's $(x,y)$ primary. Is that true?

Answer 1

1. $z\cdot z\in I$ but $z\notin I$, so $I$ is not a prime ideal.
2. By the same reason, neither $(x,y)\ $ is a prime ideal in $A$ (and $A/(x,y)\cong k[z]/(z^2)$ instead).
3. Note that elements of $A$ can be written as (represented by) polynomials $f_0\ +\ f_1\cdot z$ where $f_0,f_1\in k[x,y]$. Use this to prove that both $(x,y)$ and $I=(x,y)^2=(x^2,xy,y^2)\ $ are primary.