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Given a field, is there a functorial construction of a PID algebra over it that has infinitely many prime ideals and is not finitely generated? This excludes the ring of univariate polynomials and the ring of formal power series. I need this to construct some counterexample.

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put on hold as off-topic by Steven Landsburg, Bugs Bunny, RP_, LSpice, Sean Lawton 2 days ago

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – Steven Landsburg, Bugs Bunny, RP_, LSpice, Sean Lawton
If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ How about $k[x]$ with $x-a$ inverted for all $a \in \mathbb Z$? $\endgroup$ – Hugh Thomas Jul 11 at 21:26
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    $\begingroup$ What about K(x)[y]? $\endgroup$ – Marco Farinati Jul 11 at 22:15