---
tag: E5DH1CSW1N
kind: concept
name: Maximal order
msc: [16H10, 11R54]
specializes: [1ZBX4TKVZ9]
sources:
  - url: https://en.wikipedia.org/wiki/Order_(ring_theory)
---

## Summary

A maximal order is an [[order|1ZBX4TKVZ9]] in an algebra that is not contained in any larger order.
In an [[algebraic number field|MWBDFS2KJ5]] the ring of integers is the unique maximal order, so maximal orders generalize rings of integers to noncommutative algebras.
In noncommutative algebras maximal orders exist under mild hypotheses but are usually not unique.
For example, the conjugates of $M_2(\Z)$ by elements of $GL_2(\Q)$ are maximal orders in $M_2(\Q)$.
Maximal orders in [[quaternion algebras|ZWXW0T3EAC]] are central in the arithmetic of quaternions and of Shimura curves.

## Explanations

### L1 human

An order is a ring of "integers" inside an algebra, but an algebra can have many orders, some inside others.
For example $\Z[\sqrt{-3}]$ is an order in $\Q(\sqrt{-3})$, but it sits inside the bigger order $\Z[(1+\sqrt{-3})/2]$.
A maximal order is one that cannot be enlarged.
For number fields there is exactly one, the ring of all algebraic integers of the field.
For matrix algebras and quaternion algebras there can be several, and studying how they relate is part of the theory.