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

From Commalg

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

## Contents

## Statement

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

## 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.