# Answer to Question #22747 in Algebra for Tsit Lam

Question #22747

Show that: a commutative ring is Hilbert iff all of its quotients are rad-nil.

Expert's answer

Recall that a commutative ring

*R*iscalled Hilbert if every prime ideal in*R*is an intersection of maximalideals. Note that this property is inherited by all quotients. If*R*isHilbert, then Nil(*R*), being (always) the intersection of prime ideals in*R*, is also an intersection of maximal ideals in*R*. This shows thatNil(*R*) = rad(*R*), and the same equation also holds for allquotients of*R*. This proves the “only if” part, and the “if” part isclear.
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