# Inverse element

From Groupprops

*This is the default notion of inverse element. There is another, more general notion of inverse element in a semigroup, which does not depend on existence of a neutral element*

## Definition

### Definition with symbols

Given a set with a binary operation and a neutral element for , and given elements and we say that:

- is a
**left inverse**to if - is a
**right inverse**to if - is an
**inverse**or**two-sided inverse**to if (that is, is both a left and a right inverse to )

An element which possesses a (left/right) inverse is termed (left/right) invertible.

## Facts

### Equality of left and right inverses

If is an associative binary operation, and an element has both a left and a right inverse with respect to , then the left and right inverse are equal.

To prove this, let be an element of with left inverse and right inverse . Then, by associativity. The left side simplifies to while the right side simplifies to . Hence, .

Some easy corollaries:

- If an element has a left inverse, it can have at most one right inverse; moreover, if the right inverse exists, it must be equal to the left inverse, and is thus a two-sided inverse
- If an element has a right inverse, it can have at most one left inverse; moreover, if the left inverse exists, it must be equal to the right inverse, and is thus a two-sided inverse