---- > [!definition] Definition. ([[integral algebra]]) > Let $R$ be a [[commutative ring|commutative]] [[ring]]. We say an $R$-[[algebra]] $A$ is **integral over $R$** if every element of $A$ is $R$-[[integral element of an algebra|integral]]. > > We call an inclusion of [[ring|rings]] $A \subset B$ an **integral extension** if $B$ is integral over $A$. ^definition > [!equivalence] Equivalence for finite-type algebras. > Let $A$ be an $R$-[[algebra]]. The following are equivalent: >1. $A$ is [[subalgebra generated by a subset|finitely generated]] and integral over $R$; >2. $A$ is generated over $R$ by finitely many [[integral element of an algebra|integral elements]]; >3. $A$ is a [[finite algebra|finite]] $R$-[[algebra]]. ^equivalence > [!basicproperties] > - [[transitivity of finiteness and integrality and finite-typedness for algebras]] > - [[integral extensions, units, and fields]] > - If $A \subset B$ an integral extension, then $\frac{B}{\mathfrak{b}}$ is integral over $\frac{A}{\mathfrak{b^{}}^{c}}$ for all [[ideal|ideals]] $\mathfrak{b} \subset B$. ($\cdot^{c}$ denotes [[contraction of an ideal|contraction]]; here $\mathfrak{b}^{c}=\mathfrak{b} \cap A$.) > - The [[localization|ring of fractions]] $S ^{-1} B$ is integral over $S ^{-1} A$ for any [[multiplicative subset of a ring|multiplicative set]] $S \subset A$. > - (Localization and integral closure commute) if $\overline{A}$ is the [[integral closure]] of $A$ in $B$ and $\overline{S ^{-1} A}$ is the [[integral closure]] of $S ^{-1} A$ in $S ^{-1} B$, then $\overline{S ^{-1} A}=S ^{-1} \overline{A}$. ^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 > ```