Division ring

A division ring (or skew field) is a unital ring $D$ with $1\neq 0$ such that every nonzero element of $D$ is a unit (equivalently, $D^\times=D\setminus\{0\}$ is a group under multiplication, i.e. the group of units equals all nonzero elements).

A field is exactly a commutative division ring. Division rings occur as coefficients in noncommutative structure theorems, e.g. in the description of simple rings .

Examples:

• The Hamilton quaternions $\mathbb{H}$ form a division ring (noncommutative).
• Any field (e.g. $\mathbb{Q}$) is a division ring.
• The matrix ring $M_n(k)$ for $n\ge 2$ is not a division ring since nonzero singular matrices have no inverse.