---
tag: TBRTZMHFTV
kind: statement
role: theorem
name: Krull's height theorem
msc: ["13C15", "13E05"]
uses: [0MVH4Y1CRZ, 06HC6HK0Z8, 9BKCZPMQT6, S2D8823P8B, 030TZ0DQWT]
about: [87WH65FJB8, 0670T59HGE]
sources:
  - url: https://en.wikipedia.org/wiki/Krull%27s_principal_ideal_theorem
---

## Statement

Let $R$ be a commutative [[Noetherian ring|0MVH4Y1CRZ]] and let $I = (x_1, \dots, x_n)$ be an ideal of $R$ [[generated|030TZ0DQWT]] by $n$ elements.
Let $P$ be a [[prime ideal|06HC6HK0Z8]] of $R$ that is [[minimal|S2D8823P8B]] among the prime ideals containing $I$.
Then the [[height|9BKCZPMQT6]] of $P$ is at most $n$.

## Remarks

The proof is by induction on $n$, with [[Krull's principal ideal theorem|SJZW2HN1SC]] as the case $n = 1$.
For the step, take a prime $Q \subsetneq P$ with no prime strictly between them.
Since $P$ is minimal over $I$, some $x_i$ lies outside $Q$, and one shows that $Q$ is minimal over an ideal with $n - 1$ generators.
For example, in $k[x, y, z]$ over a field $k$, the prime $(x, y)$ is minimal over the ideal generated by $x$ and $y$, and its height is $2$.
A converse holds: every prime ideal of height $n$ in a Noetherian ring is minimal over some ideal generated by $n$ elements.
As a consequence, every prime ideal of a Noetherian ring has finite height.
A Noetherian local ring has finite [[Krull dimension|MVGBS3J1P7]], at most the number of generators of its maximal ideal.

## Explanations

### L1 human

Cutting a space by $n$ equations lowers its dimension by at most $n$.
In algebra: a prime that is minimal over an ideal with $n$ generators has height at most $n$.

### L2 human

The theorem is the base of dimension theory of Noetherian local rings.
It gives $\dim R \le$ the minimal number of generators of the maximal ideal, which leads to [[regular local rings|63XXTF6B5M]] and to [[systems of parameters|95QTEPMDA4]].
It also shows that the prime ideals of a Noetherian ring satisfy the descending chain condition.