Division ring

Definition

A division ring is a ring $D$ with these two properties.

1. $1 \neq 0$ in $D$.
2. Every nonzero element of $D$ is a unit: 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. The quaternions are a division ring that is not commutative. By Wedderburn's little theorem, 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.