---- Recall the notion of an [[integral element of an algebra]]. > [!definition] Definition. ([[integral closure]]) > Let $A \subset B$ be an inclusion of [[commutative ring|commutative]] [[ring|rings]]. The **integral closure of $A$ in $B$** is the [[subring]] $\overline{A}=\{ A\text{-}\text{integral elements of }B \}.$ We call $A$ **integrally closed in $B$** if $\overline{A}=A$. > If $A$ is in particular an [[integral domain]], then by default $B$ is assumed to be its [[field of fractions]] $\text{Frac }A$ and one speaks solely of the **integral closure of $A$** and calls $A$ **integrally closed** if $\overline{A}=A$. ^definition > [!basicexample] > - [[Every UFD is integrally closed]]. Thus, $\mathbb{Z}$ and $k[T_{1},\dots,T_{n}]$ are integrally closed. ^basic-example > [!basicproperties] > - [[taking integral closures is idempotent]] ^properties ---- #### ---- #### 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 > ```