Let $R=\Bbb{C}[x,y,z]/(xy-z^2)$ and $I=(x,z)$. Find the primary decomposition of $I^2$.
So one can verify that since $I=(x,z)$ then $I^2=(x^2,z^2,xz)$, and since $z^2=xy$ we can conclude that $I^2=(x^2,xy,xz)$.
I was able to prove that $I^2=(x^2,xy,xz)=(x)\cap(x^2,xy,z)$, and since $xy=z^2, I^2=(x^2,z^2,z)\cap(x)=(x)\cap (x^2,z)$.
Now, $(x)$ is prime in $R$ (since $(x,xy-z^2)$ is prime in $\Bbb{C}[x,y,z]$) and $(x^2,z)$ is primary and therefore this is a primary decomposition.
Now this is my first time dealing with this sort of exercise and I'm not completely sure if I'm correct. Is there anything I'm missing?
The notation you use is very confusing: $x,z$ in $I$ are residue classes of indeterminates, not indeterminates. This is why it is better to write $I=(x,z)/(xy-z^2)$. Then $I^2=\left((x,z)^2+(xy-z^2)\right)/(xy-z^2)$. But $(x,z)^2+(xy-z^2)=(x^2,xz,z^2,xy-z^2)=(x^2,xz,z^2,xy)$. A primary decomposition of the monomial ideal $(x^2,xz,z^2,xy)$ is easily found: $$(x^2,xz,z^2,xy)=(x,z^2)\cap(x^2,y,z).$$
Now $I^2=(x,z^2)/(xy-z^2)\cap(x^2,y,z)/(xy-z^2)$.