In mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element.

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.

See also:

• additive inverse
• multiplicative inverse
• group
• loop
• division ring
• unit (ring theory)