Are Noetherian commutative rings finitely generated algebras?

Question

I wonder if every Noetherian commutative ring is a finitely generated algebra over some PID.

Has this problem been proved or disproved?

Thank you for your answer.

Answer 1

Counterexamples exist. A finitely generated algebra over a PID has finite Krull dimension, but there are Noetherian rings with infinite Krull dimension.