---
tag: RVKPM94MDF
kind: concept
name: Embedded prime
msc: ["13A15", "13C05"]
specializes: [ERFZ6S8X0X]
sources:
  - url: https://en.wikipedia.org/wiki/Associated_prime
---

## Summary

An embedded prime of an ideal $I$ is an [[associated prime|ERFZ6S8X0X]] of $I$ that strictly contains another associated prime of $I$.
The associated primes that are not embedded are the minimal ones, and they are the [[minimal primes|S2D8823P8B]] over $I$.
In geometry, the minimal primes give the irreducible components of the zero set of $I$.
The embedded primes give smaller sets inside them that carry extra structure.

## Explanations

### L1 human

In the plane, the ideal $(x^2, xy)$ of $k[x, y]$ has zero set the line $x = 0$.
Its associated primes are $(x)$, for the line, and $(x, y)$, for the origin.
The origin lies inside the line, so $(x, y)$ is an embedded prime: it records a "thickening" of the line at one point that the zero set alone does not see.