54-00-KrullDimension.tex

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If $R$ is a ring, the \emph{Krull dimension} (or simply dimension) of $R$, $\dim R$ is the supremum of all integers $n$ such that there is an increasing sequence of prime ideals $\mathfrak{p}_0 \subsetneq \cdots \subsetneq \mathfrak{p}_n$ of length $n$ in $R$.

If $X$ is a topological space, the Krull dimension (or simply dimension) of $X$, $\dim X$ is the supremum of all integers $n$ such that there is a decreasing sequence of irreducible closed subsets $F_0 \supsetneq \cdots \supsetneq F_n$ of $X$.
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