Inmathematics, anidentity element orneutral element of abinary operation is an element that leaves unchanged every element when the operation is applied.[1][2] For example, 0 is an identity element of theaddition ofreal numbers. This concept is used inalgebraic structures such asgroups andrings. The termidentity element is often shortened toidentity (as in the case of additive identity and multiplicative identity)[3] when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.
Let(S, ∗) be a set S equipped with abinary operation ∗. Then an element e of S is called aleft identity ife ∗s =s for all s in S, and aright identity ifs ∗e =s for all s in S.[4] Ife is both a left identity and a right identity, then it is called atwo-sided identity, or simply anidentity.[5][6][7][8][9]
An identity with respect to addition is called anadditive identity (often denoted as 0) and an identity with respect to multiplication is called amultiplicative identity (often denoted as 1).[3] These need not be ordinary addition and multiplication—as the underlying operation could be rather arbitrary. In the case of agroup for example, the identity element is sometimes simply denoted by the symbol. The distinction between additive and multiplicative identity is used most often for sets that support both binary operations, such asrings,integral domains, andfields. The multiplicative identity is often calledunity in the latter context (a ring with unity).[10][11][12] This should not be confused with aunit in ring theory, which is any element having amultiplicative inverse. By its own definition, unity itself is necessarily a unit.[13][14]
In the exampleS = {e,f} with the equalities given,S is asemigroup. It demonstrates the possibility for(S, ∗) to have several left identities. In fact, every element can be a left identity. In a similar manner, there can be several right identities. But if there is both a right identity and a left identity, then they must be equal, resulting in a single two-sided identity.
To see this, note that ifl is a left identity andr is a right identity, thenl =l ∗r =r. In particular, there can never be more than one two-sided identity: if there were two, saye andf, thene ∗f would have to be equal to bothe andf.
It is also quite possible for(S, ∗) to haveno identity element,[15] such as the case of even integers under the multiplication operation.[3] Another common example is thecross product ofvectors, where the absence of an identity element is related to the fact that thedirection of any nonzero cross product is alwaysorthogonal to any element multiplied. That is, it is not possible to obtain a non-zero vector in the same direction as the original. Yet another example of structure without identity element involves the additivesemigroup ofpositivenatural numbers.
