----- > [!proposition] Proposition. ([[taking integral closures is idempotent]]) > Let $A \subset B$ be ([[commutative ring|commutative]]) [[ring|rings]], with $\overline{A}$ denoting the [[integral closure]] of $A$ in $B$. Then $\overline{A}$ is [[integral closure|integrally closed]] in $B$, i.e., $\overline{ \overline{A}}=\overline{A}$. > Thus, taking [[integral closure|integral closures]] is an [[idempotent]] operation. ^proposition > [!proof]- Proof. ([[taking integral closures is idempotent]]) > $\supset$. The integral closure always contains the original set. > $\subset.$ Take $x \in B$ [[integral element of an algebra|integral]] over $\overline{A}$ (i.e., $x \in \overline{ \overline{A}}$) We want to show $x \in \overline{A}$, i.e., $x$ is $A$-integral. There is a chain of inclusions $A \subset \overline{A} \subset \overline{A}[x],$ where the notation $\overline{A}[x]$ means we are [[ring adjunction|adjoining]] $x \in B$ to $A \subset B$. > Now, the extension $A \subset \overline{A}$ is [[integral algebra|integral]] by [[integral closure|definition]] of $\overline{A}$. The extension $\overline{A} \subset \overline{A}[x]$ is integral, too, [[integral algebra#^equivalence|because]] $\overline{A}[x]$ is [[subalgebra generated by a subset|generated over]] $\overline{A}$ by finitely many integral elements (namely, by $x$). By [[transitivity of finiteness and integrality and finite-typedness for algebras]], it follows that $\overline{A}[x]$ is [[integral algebra|integral]] as an [[algebra]] over $A$, and in particular that $x$ is integral over $A$. ----- #### ---- #### 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 > ```