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.
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