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Answer on Algebra Question 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 inR, 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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