Definition 10.60.1. Let $R$ be a ring. A chain of prime ideals is a sequence $\mathfrak p_0 \subset \mathfrak p_1 \subset \ldots \subset \mathfrak p_ n$ of prime ideals of $R$ such that $\mathfrak p_ i \not= \mathfrak p_{i + 1}$ for $i = 0, \ldots , n - 1$. The length of this chain of prime ideals is $n$.

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