Definition:Height (Ring Theory)

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Definition

Let $A$ be a commutative ring.

Let $\mathfrak p$ be a prime ideal in $A$.


The height of $\mathfrak p$ is the supremum over all $n$ such that there exists a chain of prime ideals:

$\mathfrak p_0 \subsetneqq \mathfrak p_1 \subsetneqq \cdots \subsetneqq \mathfrak p$