Question #22750

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

Expert's answer

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, *R*is not Hilbert.

ideal). Then

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