Definition:Inverse (Abstract Algebra)/Inverse

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 $x$.

Some sources refer to it as a reciprocal element, which terminology is borrowed from the real numbers under multiplication.

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 $x^*$.

In multiplicative notation:
 * 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

 * Definition:Left Inverse Element
 * Definition:Right Inverse Element
 * Definition:Additive Inverse
 * Definition:Multiplicative Inverse (Field Theory)