An identity element in a set is an element that is special with respect to a binary operation on the set: when an identity element is paired with any element via the operation, it returns that element.
More explicitly, let be a set and be a binary operation on Then
- an element is a left identity if for any
- an element is a right identity if for any
- an element that is both a left and right identity is called a two-sided identity, or identity element, or identity for short.
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
What are the left identities? What are the right identities?
The unique left identity is The unique right identity is also This is because the row corresponding to a left identity should read as should the column corresponding to a right identity. Note that is not a commutative operation ( and are not necessarily the same), so a left identity is not automatically a right identity (imagine the same table with the top right entry changed from to something else).
Let the set of real numbers, and let be addition. Then is an identity element: for any Note that is the unique left identity, right identity, and identity element in this case.
In general, there may be more than one left identity or right identity; there also might be none.
Let and define by the formula What are the left identities, right identities, and identity elements?
If is a left identity, then for all so so This has two solutions, so and are both left identities.
If is a right identity, then for all so so But no can be equal to for all : for instance, taking gives but taking gives This is impossible. So there are no right identities.
Because there is no element which is both a left and right identity, there is no identity element.
It is the case that if an identity element exists, it is unique:
If is a set with a binary operation, and is a left identity and is a right identity, then and there is a unique left identity, right identity, and identity element.
By the properties of identities, If is another left identity, then by the same argument, so So the left identity is unique. A similar argument shows that the right identity is unique. Since it is both a left and a right identity, so it is an identity element, and any other identity element must equal it, by the same argument.
Suppose is a set with a binary operation. Consider the following sentence about the identity elements in :
has left identities, right identities, and identity elements.
Which choice of words for the blanks gives a sentence that cannot be true?
The set of subsets of (or any set) has a binary operation given by union. The identity for this operation is the empty set since
The set of subsets of (or any set) has another binary operation given by intersection. The identity for this operation is the whole set since
Every group has a unique two-sided identity element
Every ring has two identities, the additive identity and the multiplicative identity, corresponding to the two operations in the ring. For instance, is a ring with additive identity and multiplicative identity since and for all