---
tag: 0WZXK05ME7
kind: definition
name: Division ring
msc: ["12E15"]
defines: 0EA5BTA0CX
uses: [0894J980RK, 019NFGFPR5]
sources:
  - url: https://en.wikipedia.org/wiki/Division_ring
    locator: lead section
---

## Definition

A division ring is a [[ring|0894J980RK]] $D$ with these two properties.

1. $1 \neq 0$ in $D$.
2. Every nonzero element of $D$ is a [[unit|019NFGFPR5]]: for each $a \neq 0$ in $D$, there is $b \in D$ with $a b = b a = 1$.

## Remarks

A commutative division ring is a [[field|0QA8CHH8MG]].
The [[quaternions|0GE4DVQKR8]] are a division ring that is not commutative.
By [[Wedderburn's little theorem|0RRNVX1VKA]], every finite division ring is a field.

## Explanations

### L1 human

In a division ring, you can divide by every nonzero element, as in a field.
But $a b$ and $b a$ can be different.
The quaternions of Hamilton are the standard example.