Group of Units

Theorem

Let $\left({R, +, \circ}\right)$ be a ring with unity.

Then the set $U_R$ of units of $\left({R, +, \circ}\right)$ forms a group under $\circ$.

This group $\left({U_R, \circ}\right)$ is called the group of units of the ring.

Proof

This follows directly from Invertible Elements of Monoid form Subgroup.

$\blacksquare$

Also known as

The group of units of a ring with unity $R$ is also seen denoted as $R^\times$.

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 21$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 55.5$