A division algebra $D$ is an algebra over the field $F$ such that for any element $a$ in $D$ and any non-zero element $b$ in $D$, there exists exactly one element $x\in D$ with $a=bx$ and exactly one element $y \in D$ such that $a=yb$.
If $D$ is an associative algebra, then we just require that $D$ have an identity element under the binary operation and every nonzero element have a multiplicative inverse.
Wikidata ID: Q1231309