Definition:Inverse (Abstract Algebra)/Inverse

Definition
Let $\struct {S, \circ}$ be an algebraic structure with an identity element $e_S$.

Let $x, y \in S$ be elements.

The element $y$ is an inverse of $x$ :
 * $y \circ x = e_S = x \circ y$

that is, $y$ is both:
 * a left inverse of $x$

and:
 * a right inverse of $x$.

Also see

 * Definition:Invertible Element
 * Inverse in Monoid is Unique
 * Definition:Left Inverse Element
 * Definition:Right Inverse Element


 * Definition:Additive Inverse
 * Definition:Multiplicative Inverse in Field