Answer to Question #22750 in Algebra for Tsit Lam

Question

Construct a commutative noetherian rad-nil ring that is not Hilbert.

Answer 1

Let A be a commutativenoetherian ring that is not rad-nil (e.g. the localization of Z at any maximal
ideal). Then R = A[t] is noetherian (by the Hilbert BasisTheorem), rad-nil (by Snapper’s Theorem), but its quotient R/(t) ∼ A is not rad-nil. But any Hilbert ring have to be rad-nil, Ris not Hilbert.