Answer to Question #22747 in Algebra for Tsit Lam

Question

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

Answer 1

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.