# Definition:Unit of Ring/Definition 1

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

## Contents

## Definition

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

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.

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

## Also known as

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

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 21$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 55$. Special types of ring and ring elements: $(6)$

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