Let S be a set with a binary operation *. If e is an identity element of (S,*) and a * b = e, then a is called a left inverse of b and b is called a right inverse of a. If an element x is both a left inverse and a right inverse of y, then x is called a two-sided inverse, or simply an inverse, of y. An element with a two-sided inverse is called invertible.

As with identities, it is possible for an element y to have several left inverses or several right inverses. y can even have several left inverses and several right inverses. However if the operation * is associative, then if y has both a left inverse and a right inverse, then they are equal.

An important example is the idea of an invertible square matrix. An n×n matrix M over a field K is invertible if and only if its determinant is ≠ 0. If the determinant of M is 0, it is impossible for it to have a one-sided inverse; therefore a left inverse or right inverse implies the existence of the other one.