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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.

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  • $\begingroup$ What does the notation $P$ in $P(|Specm(R)|)$ mean, actually? $\endgroup$ – Watson Jan 26 '17 at 13:32
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    $\begingroup$ @Watson It is meant to denote the powerset $\endgroup$ – Pax Kivimae Jan 26 '17 at 22:28
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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.

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  • $\begingroup$ I can't believe I didn't even check if for local rings. I think I might be a bit off today. Thank you so much! $\endgroup$ – Pax Kivimae Jan 19 '17 at 20:44
  • $\begingroup$ You can have other examples here or on this website (from this user). $\endgroup$ – Watson Jan 21 '17 at 11:42
  • $\begingroup$ @watson Hi: thanks for the suggestion! When I checked, I did not see another such ring in storage, so it is valuable indeed. I'd like to help flesh it out more, but I need a second opinion. It is UFD that is not a PID, is that correct? It seems like $(X,Y)$ should not be principal, but I just want to run it by another person. $\endgroup$ – rschwieb Jan 24 '17 at 19:26
  • $\begingroup$ @rschwieb : you're welcome :-). I agree that, as a localization of a UFD, it is a UFD, and it has dimension $2$, so it is not a PID. [By the way, it would be so great to have some property giving the Krull dimension of the rings at ringtheory.herokuapp.com – for instance there is an amazing example of Noetherian ring with infinite Krull dimension…] $\endgroup$ – Watson Jan 24 '17 at 19:43
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    $\begingroup$ @Watson I will think about adding it as something visible for commutative rings. As a substitute, I have been inserting it into the ring's extra info. But it would be nice to be able to search on Krull dimension for commutative rings :) $\endgroup$ – rschwieb Jan 24 '17 at 19:54

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