I`m struggling with the following solution I wanted to use:
Let S := $\mathbb{C}[x,y,z] /(z^2-xy) \cong \mathbb{C}[x,y,\sqrt{xy}]$
Then $S/(x-y) \cong \mathbb{C}[x,x,\sqrt{x^2}] = \mathbb{C}[x]$, which is an integral domain.
Hence $(x-y)$ is prime in S, but $$xy-y^2=y(x-y) \in (x-y)$$ and $$xy-y^2 = (\sqrt{xy}-y)(\sqrt{xy}+y),$$ where neither of the last two factors is in $(x-y)$, which means $(x-y)$ is not prime.
Where did I go wrong?