Does there exist such an affine scheme?

Question

I aim to find out an affine scheme $X$ such that satisfies:

1 $X$ is irreducible.

2 $\exists f\in\mathcal O(X)$ such that $X\setminus Z_f$ is neither irreducible nor empty.

I have stuck on it for quite a while so that I wonder if such an example exists...

If such an $X$ exists, may I please ask for some examples? Is there any way to construct such an example? Or if it does not exist, may I please ask why?

Thanks for any answers, hints or references.

Answer 1

Irreducible means " every two non-void open subsets intersect", that is "every non-void open subset is dense" . From this, one sees right away that this passes down to open subsets: every open non-void subspace of an irreducible subspace is irreducible.

Therefore, you can't find such an example