Definition:Height of Proper Ideal

Definition

Let $A$ be a commutative ring with unity.

Let $I$ be a proper ideal in $A$.

The height of $I$ is defined as:

$\map {\operatorname {ht} } I := \inf \set {\map {\operatorname {ht} } {\mathfrak p} : \mathfrak p \in \Spec A \text{ s.t. } I \subseteq \mathfrak p }$

where:

$\map {\operatorname {ht} } {\mathfrak p}$ is the height of $\mathfrak p$

$\Spec A$ is the prime spectrum of $A$

Sources

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