# Definition:Inverse (Abstract Algebra)

## Definition

Let $\struct {S, \circ}$ be a monoid whose identity is $e_S$.

### Left Inverse

An element $x_L \in S$ is called a left inverse of $x$ if and only if:

$x_L \circ x = e_S$

### Right Inverse

An element $x_R \in S$ is called a right inverse of $x$ if and only if:

$x \circ x_R = e_S$

### Inverse

The element $y$ is an inverse of $x$ if and only if:

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

that is, if and only if $y$ is both:

a left inverse of $x$

and:

a right inverse of $x$.