Noetherian ring has finitely many minimal primes and every prime contains a minimal prime

Statement
In a fact about::Noetherian ring the following are true:


 * Every fact about::prime ideal contains a fact about::minimal prime ideal
 * There are only finitely many minimal prime ideals
 * The fact about::nilradical is the intersection of that finite collection of minimal primes

Facts used
This result is best proved using the spectrum and the following facts about it:


 * The spectrum of a Noetherian ring is a Noetherian space.
 * The minimal prime ideals of a ring correspond to the maximal irreducible closed subsets of its spectrum.
 * A Noetherian space has only finitely many maximal irreducible closed subsets, called its irreducible components, and is the union of those.