Definition:Height of Proper Ideal
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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