----- > [!proposition] Proposition. ([[being an integrally closed domain is a local property]]) > Let $A$ be an [[integral domain]]. The following are equivalent: > 1. $A$ is [[integral closure|integrally closed]]; > 2. $A_{\mathfrak{p}}$ is [[integral closure|integrally closed]] for all [[prime ideal|primes]] $\mathfrak{p} \in \text{Spec }A$; > 3. $A_{\mathfrak{m}}$ is [[integral closure|integrally closed]] for all [[maximal ideal|maximals]] $\mathfrak{m} \in \text{mSpec }A$. > [!proof]- Proof. ([[being an integrally closed domain is a local property]]) > ~ ----- #### - [[localization]], [[localization functor]] ---- #### 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 > ```