I learnd that the maximal ideals in $\mathbb C[x,y]$ have the form $(x-z\_1, y-z\_2)$ by the Nullstellensatz. But if we set $I=(xy-1)$ then $\mathbb C[x,y]/I$ is isomorphic to $\mathbb C[x,1/x]$ which is in my opinion a field, thus $(xy-1)$ is maximal. Where did I make a mistake?