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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 ringIn abstract algebra, a division ring also called a skew field is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. an element x with ax xa 1). Division rings are very similar to fields except that their multiplic
• unit (ring theory)Ring theory In mathematics, a unit in a ring R is an element u such that there is v in R with uv vu 1. That is, u is an invertible element of the multiplicative monoid of R''. The units of R form a group U ''R under multiplication, the group of units of R