Definition 10.60.2. The Krull dimension of the ring $R$ is the Krull dimension of the topological space $\mathop{\mathrm{Spec}}(R)$, see Topology, Definition 5.10.1. In other words it is the supremum of the integers $n\geq 0$ such that $R$ has a chain of prime ideals
\[ \mathfrak p_0 \subset \mathfrak p_1 \subset \ldots \subset \mathfrak p_ n, \quad \mathfrak p_ i \not= \mathfrak p_{i + 1}. \]
of length $n$.
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