## You are here

Homeheight of a prime ideal

## Primary tabs

# height of a prime ideal

Let $R$ be a commutative ring and $\mathfrak{p}$ a prime ideal of $R$. The height of $\mathfrak{p}$ is the supremum of all integers $n$ such that there exists a chain

$\mathfrak{p}_{0}\subset\cdots\subset\mathfrak{p}_{n}=\mathfrak{p}$ |

of distinct prime ideals. The height of $\mathfrak{p}$ is denoted by $\operatorname{h}(\mathfrak{p})$.

$\operatorname{h}(\mathfrak{p})$ is also known as the rank of $\mathfrak{p}$ and the codimension of $\mathfrak{p}$.

The Krull dimension of $R$ is the supremum of the heights of all the prime ideals of $R$:

$\sup\{\operatorname{h}(\mathfrak{p})\mid\mathfrak{p}\mbox{ prime in }R\}.$ |

Defines:

rank of an ideal, codimension of an ideal

Related:

KrullDimension, Cevian

Synonym:

height

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

14A99*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections