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\}.

Title	height of a prime ideal
Canonical name	HeightOfAPrimeIdeal
Date of creation	2013-03-22 12:49:25
Last modified on	2013-03-22 12:49:25
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	10
Author	CWoo (3771)
Entry type	Definition
Classification	msc 14A99
Synonym	height
Related topic	KrullDimension
Related topic	Cevian
Defines	rank of an ideal
Defines	codimension of an ideal