---
tag: 1B4G153ZGE
kind: definition
name: Krull dimension
msc: ["13C15"]
defines: MVGBS3J1P7
uses: [0GYGHKX834, 06HC6HK0Z8]
sources:
  - url: https://en.wikipedia.org/wiki/Krull_dimension
---

## Definition

Let $R$ be a nonzero [[commutative ring|0GYGHKX834]].
A chain of [[prime ideals|06HC6HK0Z8]] $P_0 \subsetneq P_1 \subsetneq \cdots \subsetneq P_n$ of $R$ has length $n$.
The *Krull dimension* $\dim R$ is the supremum of the lengths of all such chains, an element of $\set{0, 1, 2, \dots} \cup \set{\infty}$.

## Remarks

For the zero ring, which has no prime ideals, the dimension is set to $-1$ or $-\infty$ by convention.
Example: $\dim \Z = 1$; the chain $(0) \subsetneq (p)$ for a prime $p$ has length $1$, and no chain is longer, since every nonzero prime ideal of $\Z$ is maximal.
Non-example of a chain: $(0) \subsetneq (6) \subsetneq (3)$ does not show $\dim \Z \ge 2$, because $(6)$ is not a prime ideal.
For a field $k$, $\dim k[x_1, \dots, x_n] = n$, and the polynomial ring $k[x_1, x_2, \dots]$ in infinitely many variables has infinite dimension.
A [[Noetherian ring|0MVH4Y1CRZ]] can still have infinite dimension, by [[an example of Nagata|YEJVKCY78E]].

## Explanations

### L1 human

The Krull dimension of a ring is the length of the longest strictly increasing chain of prime ideals.