# A problem involving prime ideals that are not maximal in $K[x,y]$

Let a non-zero ideal $$P \in \operatorname{Spec}(K[x,y])$$ be a non-maximal ideal, where $$K$$ is an algebraically closed field. Can I say that $$P \cap K[x] = (0)$$ or $$P \cap K[y] = (0)$$?

If not, can you think of any counterexamples? Any help is welcome. Thanks!

• What have you tried so far? Can you please write down a few principal ideals and check? Jun 28 at 0:38
• You mean a, not the, right? Jun 28 at 0:43
• It's "a" instead of "the". Sorry! Jun 28 at 0:50

Let us suppose $$K[x] \cap P$$ is nontrivial. Since $$K$$ is algebraically closed and $$P$$ is prime, there is some $$a \in K$$ such that $$x - a \in P$$.
And if $$K[y] \cap P$$ is also nontrivial, then there is some $$b \in K$$ such that $$y - b \in P$$.
Let $$R = K[x, y] / (x - a, y - b) \cong K$$, and let $$\pi : K[x, y] \to R$$ be the canonical map. Then $$R / \pi(P) \cong K[x, y]/P$$.
Now since $$P$$ is prime, we see that $$K[x, y] / P \cong R/\pi(P)$$ is an integral domain. Therefore, $$\pi(P)$$ is prime, hence maximal. Then $$P$$ is maximal.