# Infinitely generated PID algebra with infinitely many prime ideals [on hold]

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.

## put on hold as off-topic by Steven Landsburg, Bugs Bunny, RP_, LSpice, Sean Lawton2 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.

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