---
tag: 4K6F157663
kind: definition
name: Integral extension
msc: [13B21]
defines: B7X0KFY1S1
uses: [1EHN4KPSFS, 0GNCRPYXYV, 0GGE0Z0JQ1]
sources:
  - url: https://en.wikipedia.org/wiki/Integral_element
---

## Definition

Let $B$ be a commutative ring and let $A$ be a [[subring|0GNCRPYXYV]] of $B$.
The ring $B$ is *integral over $A$*, and $A \subseteq B$ is an *integral extension*, if every element of $B$ is [[integral|1EHN4KPSFS]] over $A$.
A [[ring homomorphism|0GGE0Z0JQ1]] $f \colon A \to B$ of commutative rings is *integral* if $B$ is integral over the subring $f(A)$.

## Remarks

Example: $\Z \subseteq \Z[i]$ is integral.
The element $a + bi$ with $a, b \in \Z$ is a root of the monic polynomial $x^2 - 2a x + a^2 + b^2$ with integer coefficients.

Non-example: $\Z \subseteq \Q$ is not integral, because $1/2$ is not integral over $\Z$.

An extension is integral and of finite type if and only if it is [[finite|1NBACAEC2V]].
Integrality is [[transitive|ZXZMQM818D]], and it is kept by quotients and localizations.
If $B$ is integral over $A$, then $B/J$ is integral over $A/(J \cap A)$ for every ideal $J$ of $B$.
Also $S^{-1}B$ is integral over $S^{-1}A$ for every multiplicative set $S$ of $A$.