----- - [[prime ideal]] - [[height of a prime ideal]] > [!proposition] Proposition. ([[Algebraic Hartog's Lemma]]) > If $A$ is an [[integral closure|integrally closed]] [[Noetherian ring|Noetherian]] [[integral domain]], then $A=\bigcap_{\text{ht }\mathfrak{p}=1, \mathfrak{p} \subset A \text{ prime }}^{}A_{\mathfrak{p}} \subset A_{(0)}.$ ^proposition > [!proof]- Proof. ([[Algebraic Hartog's Lemma]]) > Won't prove in our course. See Matsumara Theorem 38 (Page 124). ----- #### ---- #### 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 > ```