---- > [!definition] Definition. ([[integrality over an ideal]]) > Let $A \subset B$ be an extension of ([[commutative ring|commutative]]) [[ring|rings]], $\mathfrak{a}$ an [[ideal]] of $A$. > > We say that $x \in B$ is **integral over the ideal $\mathfrak{a} \subset A$** (or **$\mathfrak{a}$-integral**) if there is a [[monic polynomial|monic]] [[polynomial 4|polynomial]] $f=T^{n}+a_{1}T^{n-1}+\dots+a_{n}T^{0} \in A[T]$, $a_{i} \in \mathfrak{a}$, such that $f(x)=0$. > > The **integral closure** of $\mathfrak{a}$ in $B$ is $\{ x \in B : x \text{ is } \mathfrak{a}\text{-integral}\}$. It is an [[ideal]] of $B$ (in particular, it is stable under addition and multiplication), [[integral closure of an ideal is the radical of its extension|because it equals]] the [[radical of an ideal|radical]] $\sqrt{ \mathfrak{a} \overline{A} }$, where $\overline{A}$ denotes the [[integral closure]] of $A$ in $B$. > > [!equivalence] > - [[characterization of integrality over an ideal (in an integrally closed domain)]] ^equivalence ---- #### ---- #### 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 > ```