Show that: a commutative ring is Hilbert iff all of its quotients are rad-nil.
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.
Thank you for the best service that you guys give to us. I can't thank you enough for helping me with my work and assignment. You guys make sure to have quality over quantity and I can really see that! Thanks again.