# Answer to Question #22750 in Algebra for Tsit Lam

Question #22750

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

Expert's answer

Let

ideal). Then

*A*be a commutativenoetherian ring that is not rad-nil (e.g. the localization of Z at any maximalideal). 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,*R*is not Hilbert.
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