I'm getting a contradiction with mathematics. Can you find my mistake?
So, let $A = \mathbb{C}[x,y]/(xy)$. Then $\mathfrak{p}=(x-1,y)$ is a prime ideal in $A$, because it's a prime ideal in $\mathbb{C}[x,y]$ containing $(xy)$. Let $A_{\mathfrak{p}}$ be the localization at $\mathfrak{p}$ (i.e. we invert all the elements that aren't in $\mathfrak{p}$).
Then, $x$ is a unit in $A_{\mathfrak{p}}$, because it wasn't in the ideal. But also $xy=0$, so $x$ is also a zero-divisor. This is absurd.
I must be missing something obvious, but I can't see what.
There is no contradiction since $y=0$ in $A_{\mathfrak p}$. In fact, it is better to write $\frac y1=\frac 01$, and this holds since there is an element in $A\setminus\mathfrak p$ which annihilates $y$ (guess which one?).