Group of units of a ring

Definition

Let $R$ be a ring. The group of units of $R$ is the set $R^\times$ of the units of $R$, with the multiplication of $R$. It is a group with identity element $1$.

Remarks

The set $R^\times$ is closed under multiplication: if $u$ and $v$ are units, then $(u v)^{-1} = v^{-1} u^{-1}$. Other notations for $R^\times$ are $R^*$ and $U(R)$. The group $\mathbb{Z}^\times = \{1, -1\}$ has order $2$. For a field $K$, $K^\times = K \setminus \{0\}$.

Explanations

L1 human

The units of a ring are the elements with an inverse. If you multiply two units, you get a unit again. So the units form a group under multiplication.