More explicitly, let be a set, a binary operation on and Suppose that there is an identity element for the operation. Then
- an element is a left inverse for if
- an element is a right inverse for if
- an element is an inverse (or two-sided inverse) for if it is both a left and right inverse for
Let and consider the binary operation defined by the following table: The value of is given by looking up the row with and the column with
Which elements have left inverses? Right inverses? Inverses?
Since is the identity, and it follows that
- has no left or right inverse;
- has no left inverse, and has a right inverse
- has a left inverse and a right inverse
- is its own left and right inverses.
Consider the set with the binary operation of addition. The identity element is so the inverse of any element is as So every element has a unique left inverse, right inverse, and inverse.
An element might have no left or right inverse, or it might have different left and right inverses, or it might have more than one of each.
Let be the set of functions There is a binary operation given by composition i.e. The (two-sided) identity is the identity function
Find a function with more than one left inverse. Find a function with more than one right inverse.
If then has more than one left inverse: let and let Then and because is always positive. The idea is that and are the same on positive values, which are in the range of but differ on negative values, which are not.
If then has more than one right inverse: let and Then
If the binary operation is associative and has an identity, then left inverses and right inverses coincide:
If is a set with an associative binary operation with an identity element, and an element has a left inverse and a right inverse then and has a unique left, right, and two-sided inverse.
Let be the identity. Then The same argument shows that any other left inverse must equal and hence Similarly, any other right inverse equals and hence So there is exactly one left inverse and exactly one right inverse, and they coincide, so there is exactly one two-sided inverse.
The existence of inverses is an important question for most binary operations. Here are some examples.
Let with It is straightforward to check that this is an associative binary operation with two-sided identity Then the inverse of if it exists, is the solution to which is but when this inverse does not exist; indeed for all So every element of has a two-sided inverse, except for
Let be the set of sequences where the are real numbers. Let be the set of functions Then composition of functions is an associative binary operation on with two-sided identity given by the identity function. Now let be the shift operator, Then has many left inverses but no right inverses (because is injective but not surjective). One of its left inverses is the reverse shift operator
Let be a group. Then every element of the group has a two-sided inverse, even if the group is nonabelian (i.e. the operation is not commutative).
Let be a ring. Then every element of has a two-sided additive inverse is a group under addition but not every element of has a multiplicative inverse. In particular, never has a multiplicative inverse, because for all If every other element has a multiplicative inverse, then is called a division ring, and if is also commutative, then it is called a field. In general, the set of elements of with two-sided multiplicative inverses is called the group of units of