Noetherian ring has finitely many minimal primes and every prime contains a minimal prime
This article defines a result where the base ring (or one or more of the rings involved) is Noetherian
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In a Noetherian ring the following are true:
- Every prime ideal contains a minimal prime ideal
- There are only finitely many minimal prime ideals
- The nilradical is the intersection of that finite collection of minimal primes
Further information: Noetherian ring
Further information: Prime ideal
Minimal prime ideal
Further information: Minimal prime ideal
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.