---- > [!definition] Definition. ([[height of an ideal]]) > Let $R$ be a ([[commutative ring|commutative]]) [[ring]] and $I \subset R$ any [[ideal]]. The **height** of $I$ [[height of a prime ideal|is]] $\text{ht }I=\inf\{ \text{ht }\mathfrak{p}: \mathfrak{p} \supset I \text{ with } \mathfrak{p} \text{ prime} \}.$ > [[codimension of a closed subspace|Equivalently]] $\text{ht }\mathfrak{p}=\text{codim}(V(I), \text{Spec }A)$ where the [[codimension of a closed subspace|codimension]] is that of a [[closed set|closed]] [[subspace topology|subspace]] of the [[topological space|topological]] [[Zariski topology on a ring spectrum|space]] $\text{Spec }A$. ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```