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

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This article defines a result where the base ring (or one or more of the rings involved) is Noetherian
View more results involving Noetherianness or Read a survey article on applying Noetherianness

Statement

In a Noetherian ring the following are true:

Definitions used

Noetherian ring

Further information: Noetherian ring

Prime ideal

Further information: Prime ideal

Minimal prime ideal

Further information: Minimal prime ideal

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.