# Definition:Unit of Ring

*Not to be confused with Definition:Unity of Ring.*

## Contents

## Definition

Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$.

### Definition 1

An element $x \in R$ is a **unit of $\struct {R, +, \circ}$** if and only if $x$ is invertible under $\circ$.

That is, a **unit** of $R$ is an element of $R$ which has an inverse.

- $\exists y \in R: x \circ y = 1_R = y \circ x$

### Definition 2

An element $x \in R$ is a **unit of $\struct {R, +, \circ}$** if and only if $x$ is divisor of $1_R$.

### Product Inverse

The inverse of $x \in U_R$ by $\circ$ is called the **(ring) product inverse of $x$**.

The usual means of denoting the product inverse of an element $x$ is by $x^{-1}$.

Thus it is distinguished from the additive inverse of $x$, that is, the (ring) negative of $x$, which is usually denoted $-x$.

### Group of Units

Let $\struct {R, +, \circ}$ be a ring with unity.

Then the set $U_R$ of units of $\struct {R, +, \circ}$ is called the **group of units** of $\struct {R, +, \circ}$.

This can be denoted explicitly as $\struct {U_R, \circ}$.

## Also known as

Some sources use the term **invertible element** for **unit of ring**.

## Also see

- Results about
**units of rings**can be found here.

## Sources

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 6$: Rings and fields