Definition:Height of Prime Ideal

Definition

Let $A$ be a commutative ring with unity.

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:

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

It is denoted by:

$\map {\operatorname {ht} } {\mathfrak p}$

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Sources

• 1980: Hideyuki Matsumura: Commutative Algebra $12:$ Dimension