 Unit group

The units of a ring $R$ form a group under the ring's multiplication, denoted $R^{\times }$. Closure follows because a product of units has inverseThe identity element is the ring's one, associativity is inherited, and each multiplicative inverse is again a unit. The reversed order in the displayed formula is essential in a noncommutative ring.

For example, ${\mathbb{Z}}^{\times }=\{1,-1\}$ and $(\mathbb{Z}/8\mathbb{Z}{)}^{\times }=\{1,3,5,7\}$. In the second group, each nonidentity element has square one. For a field $k$, the unit group of the matrix ring $M_n(k)$ is the general linear group ${\mathrm{GL}}_n(k)$, connecting the abstract definition with invertible matrices.