---
tag: 019NFGFPR5
kind: definition
name: Unit of a ring
msc: [16U60]
defines: 0XVEZDE9J8
uses: [0894J980RK]
sources:
  - url: https://en.wikipedia.org/wiki/Unit_(ring_theory)
    locator: lead section
---

## Definition

Let $R$ be a [[ring|0894J980RK]].
An element $u \in R$ is a unit of $R$ if there is an element $v \in R$ such that $u \cdot v = v \cdot u = 1$.
The element $v$ is unique. It is the inverse of $u$, with the notation $u^{-1}$.

## Remarks

The units of $\mathbb{Z}$ are $1$ and $-1$.
In a [[field|0QA8CHH8MG]], every nonzero element is a unit.
A real $n \times n$ matrix is a unit of the matrix ring if and only if its determinant is not zero.
The units of $R$ form the [[group of units|05THZKR9AN]] of $R$.

## Explanations

### L1 human

A unit is an element that you can divide by.
In the integers, you can divide by $1$ and $-1$ and stay in the integers, but you cannot divide by $2$.
So $1$ and $-1$ are the only units of $\mathbb{Z}$.