# Definition:Inverse (Abstract Algebra)/Inverse

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< Definition:Inverse (Abstract Algebra)(Redirected from Definition:Inverse Element)

## Contents

## Definition

Let $\left({S, \circ}\right)$ be an algebraic structure with an identity element is $e_S$.

Let an element $y \in S$ be such that:

- $y \circ x = e_S = x \circ y$

that is, $y$ is both:

- a left inverse of $x$

and:

- a right inverse of $x$.

Then $y$ is an **inverse of $x$**.

## Also known as

An **inverse of $x$** can also be referred to as a **two-sided inverse of $x$**.

The notation used to represent an inverse of an element depends on the set and binary operation under consideration.

Various symbols are seen for a general inverse, for example $\hat x$ and $x^*$.

- If $s \in S$ has an inverse, it is denoted $s^{-1}$.

If the operation concerned is commutative, then additive notation is often used:

- If $s \in S$ has an inverse, it is denoted $-s$.

## Also see

- Results about
**Inverse Elements**can be found here.

## Sources

- Iain T. Adamson:
*Introduction to Field Theory*(1964)... (previous)... (next): $\S 1.1$ - J.A. Green:
*Sets and Groups*(1965)... (previous)... (next): $\S 4.4$ - Seth Warner:
*Modern Algebra*(1965)... (previous)... (next): $\S 4$ - George McCarty:
*Topology: An Introduction with Application to Topological Groups*(1967)... (previous)... (next): $\text{II}$: The Group Property - B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*(1970)... (previous)... (next): $\S 1.1$: The definition of a ring: Definitions $1.1 \ \text{(b)}$ - Allan Clark:
*Elements of Abstract Algebra*(1971)... (previous)... (next): $\S 27$ - A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*(1972)... (previous)... (next): $\S 5$: Groups $\text{I}$ - Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*(1978)... (previous)... (next): $\S 31$ - P.M. Cohn:
*Algebra Volume 1*(2nd ed., 1982)... (previous)... (next): $\S 3.1$: Monoids