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.
1
put on hold as offtopic by Steven Landsburg, Bugs Bunny, RP_, LSpice, Sean Lawton 2 days ago
This question appears to be offtopic. 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

1$\begingroup$ How about $k[x]$ with $xa$ inverted for all $a \in \mathbb Z$? $\endgroup$ – Hugh Thomas Jul 11 at 21:26

1$\begingroup$ What about K(x)[y]? $\endgroup$ – Marco Farinati Jul 11 at 22:15