# Definition:Inverse (Abstract Algebra)/Inverse

(Redirected from Definition:Inverse Element)

## 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.