inverse element

An inverse element is a generalization of the concepts of opposite ($-x$) and reciprocal ($\frac{1}{x}$).

Given an operation $\ast$ and an identity element $e$:

If $x\ast y=e$, we say $x$ is the left inverse of $y$ and that $y$ is the right inverse of $x$.

If $\ast$ is associative, and if element $x$ has both a left inverse and right inverse, then these two inverses are equal and unique; they are called the inverse element or simply the inverse.

Inverses are commonly used in algebraic groups, where every element is invertible, and rings, where invertible elements are also called units.