---
tag: 1ZBX4TKVZ9
kind: concept
name: Order (ring theory)
msc: [16H10, 11R54, 16G30]
specializes: [0GNCRPYXYV]
sources:
  - url: https://en.wikipedia.org/wiki/Order_(ring_theory)
same_as:
  wikidata: Q1431456
  wikipedia: Order (ring theory)
---

## Summary

An order is a [[subring|0GNCRPYXYV]] of a finite-dimensional algebra over a field.
It is a finitely generated module over a ring of integers such as $\Z$, and it spans the algebra over the field.
The basic examples are the subrings of an [[algebraic number field|MWBDFS2KJ5]] that are free of full rank over $\Z$, such as $\Z[\sqrt{-3}]$ in $\Q(\sqrt{-3})$.
Orders in noncommutative algebras include $M_n(\Z)$ in $M_n(\Q)$ and the integral group ring $\Z G$ in $\Q G$.
They are central in integral representation theory and in the arithmetic of quaternion algebras.
The largest orders, the [[maximal orders|E5DH1CSW1N]], generalize rings of integers.

## Explanations

### L1 human

The integers sit inside the rationals as a grid of points that still spans everything once fractions are allowed.
An order is the same idea for a bigger algebra: a subring that is a grid over $\Z$ and whose rational multiples fill the whole algebra.
The Gaussian integers $\Z[i]$ form an order in $\Q(i)$, with grid basis $1, i$.
The integer matrices form an order in the rational matrices.
Orders let one do number theory, with divisibility and ideals, inside algebras that are not fields.