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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