[[Ring theory MOC]]
# Krull dimension

Let $R$ be a [[commutative ring]] and $\mathfrak{p} \triangleleft R$ be a [[prime ideal]].
The **height** $\opn{ht} \mathfrak{p}$ of $\mathfrak{p}$ is the supremum of the lengths of strictly increasing sequences of prime ideals culminating in $\mathfrak{p}$
$$
\begin{align*}
\mathfrak{p}_{0} \subsetneq \mathfrak{p}_{1}  \subsetneq \dots \subsetneq \mathfrak{p}_{\opn{ht} \mathfrak{p}} = \mathfrak{p}
\end{align*}
$$
and the **Krull dimension** $\dim R$ of $R$ is the supremum of the height of all prime ideals.

## Properties

- [[Krull dimension of an integral domain]]

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