---
tag: 87WH65FJB8
kind: concept
name: Height of a prime ideal
msc: ["13C15"]
sources:
  - url: https://en.wikipedia.org/wiki/Krull_dimension
---

## Summary

The height of a prime ideal $P$ is the largest length of a strictly increasing chain of prime ideals that ends at $P$.
It equals the [[Krull dimension|MVGBS3J1P7]] of the [[localization|X6JD4AG973]] at $P$, and it is the algebraic notion of codimension.
[[Krull's principal ideal theorem|SJZW2HN1SC]] and [[Krull's height theorem|TBRTZMHFTV]] bound the height by the number of generators of an ideal.

## Explanations

### L1 human

The dimension of a ring counts the longest staircase of prime ideals anywhere.
The height of a prime $P$ counts only the staircases that end at $P$, looking down from $P$.
In $k[x, y, z]$, the prime $(x)$ is a surface and has height $1$, while $(x, y)$ is a line and has height $2$: it has codimension $2$ in space.