Let be a ring with unity and . Then is left invertible if there exists with ; is a left
inverse of . Similarly, is right invertible if there exists with ; is a right inverse of .
Note that, if is left invertible, it may not have a unique left inverse, and similarly for right invertible elements. On the other hand, if is left invertible and right invertible, then it has exactly one left inverse and one right inverse. Moreover, these two inverses are equal, and is a unit.