This question comes from the naive belief that $|Spec(R)|\subset P(|Specm(R)|)$, which I now know is only true if $R$ is a Jacobson ring, which lead me to believe that Semilocal rings are characterized by having a finite number of prime ideals, rather than just maximal. I now believe this is false, but I cannot come up with a counterexample. Any help would be greatly appreciated.
2026-03-26 20:41:13.1774557673
What is an example of a Noetherian Semi-local ring with an infinite number of prime ideals?
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You can try the local ring $\Bbb Q[X,Y]_{(X,Y)}$, which is noetherian (as localization of a noetherian ring — $\Bbb Q[X,Y]$ is noetherian by the Hilbert basis theorem). The ideals $(X+aY)$ in $\Bbb Q[X,Y]_{(X,Y)}$ are prime and pairwise distinct, so that the spectrum of $\Bbb Q[X,Y]_{(X,Y)}$ is infinite.