Suppose we have a binary operation on a set , , and suppose this operation has an identity , so that for every we have . An inverse to under this operation is an element such that .
Thus, informally, operating by is the "opposite" of operating by -inverse.
If our operation is not commutative, we can talk separately about left inverses and right inverses. A left inverse of would be some such that , while a right inverse would be some such that .
Uniqueness (under appropriate conditions)
If the operation is associative and an element has both a right and left inverse, these two inverses are equal.
Let be the element with left inverse and right inverse , so . Then , by the properties of . But by associativity, , so we do indeed have .
If the operation is associative, inverses are unique.