---
tag: 05THZKR9AN
kind: definition
name: Group of units of a ring
msc: [16U60]
defines: 07C6ECNJET
uses: [019NFGFPR5, G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Unit_(ring_theory)
    locator: Group of units
---

## Definition

Let $R$ be a [[ring|0894J980RK]].
The group of units of $R$ is the set $R^\times$ of the [[units|019NFGFPR5]] of $R$, with the multiplication of $R$.
It is a [[group|G5S1M9]] 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.