# height of a prime ideal

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Let $R$ be a commutative ring and $\mathfrak{p}$ a prime ideal of $R$.  The {\em 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 $R$: $$\sup\lbrace \operatorname{h}(\mathfrak{p}) \mid \mathfrak{p}\mbox{ prime in }R \rbrace.$$
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