---
tag: S2D8823P8B
kind: concept
name: Minimal prime ideal
msc: [13A15]
specializes: [00MHHZ9SNE]
sources:
  - url: https://en.wikipedia.org/wiki/Minimal_prime_ideal
same_as:
  wikidata: Q17098959
  wikipedia: Minimal prime ideal
---

## Summary

A minimal prime ideal of a commutative ring is a [[prime ideal|06HC6HK0Z8]] that contains no smaller prime ideal.
Every prime ideal contains a minimal prime, and the [[nilradical|QQDN6EYQNJ]] is the intersection of the minimal primes.
Geometrically, the minimal primes correspond to the irreducible components of the [[spectrum|FB45BHHA2J]].
A Noetherian ring has only finitely many minimal primes.

## Explanations

### L1 human

Prime ideals can be ordered by inclusion, and the minimal ones sit at the bottom.
In an integral domain such as $\Z$, the zero ideal is prime, so it is the only minimal prime.
In the ring $k[x, y]/(xy)$, which describes the union of the two coordinate axes, there are two minimal primes, generated by $x$ and by $y$.
They correspond to the two axes, the two irreducible pieces of the picture.