Group of units

The multiplicative group consisting of all units in a unital ring.

Group of units

For a unital ring $R$ , the group of units is

R^\times=\{u\in R: u \text{ is a unit}\}

with operation given by multiplication in $R$ . This is a group , and it consists exactly of the units of $R$ .

The unit group is functorial: a unital ring homomorphism sends units to units. In noncommutative settings, unit groups encode significant structure (e.g. general linear groups).

Examples:

$\mathbb Z^\times=\{\pm 1\}$ .
$(\mathbb Z/n\mathbb Z)^\times$ is the group of residue classes coprime to $n$ .
$M_n(k)^\times$ is the group $\mathrm{GL}_n(k)$ of invertible matrices.