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 HeightOfAPrimeIdeal 2013-03-22 12:49:25 2013-03-22 12:49:25 CWoo (3771) CWoo (3771) 10 CWoo (3771) Definition msc 14A99 height KrullDimension Cevian rank of an ideal codimension of an ideal