---
tag: B7X0KFY1S1
kind: concept
name: Integral extension
msc: [13B21]
sources:
  - url: https://en.wikipedia.org/wiki/Integral_element
---

## Summary

A ring extension $A \subseteq B$ is integral if every element of $B$ is [[integral|1EHN4KPSFS]] over $A$.
It is the ring analogue of an [[algebraic extension|0PTBM7K0R2]] of fields.
Integral extensions keep the prime ideals close to those of the base.
The lying over, going-up and incomparability theorems hold, and the two rings have the same Krull dimension.
Every [[finite|1NBACAEC2V]] ring extension is integral.
In geometry an integral extension gives a closed map of spectra whose fibres have dimension zero.

## Explanations

### L1 human

An integral extension adds to a ring only elements that satisfy monic equations over it.
The Gaussian integers $\Z[i]$ are an integral extension of $\Z$, since every $a + bi$ is a root of $x^2 - 2a x + (a^2 + b^2)$.
The rational numbers are not an integral extension of $\Z$, since $1/2$ is not integral.
A picture: the parabola $y^2 = x$ projected to the $x$-axis corresponds to $k[x] \subseteq k[x, y]/(y^2 - x)$, an integral extension, and every point of the axis has one or two points above it.