Identity element

In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them. This concept is used in algebraic structures such as groups. The term identity element is often shortened to identity (as will be done in this article) when there is no possibility of confusion.

Let (S, ∗) be a set S with a binary operation ∗ on it. Then an element e of S is called a left identity if e ∗ a = a for all a in S, and a right identity if a ∗ e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.

An identity with respect to addition is called an additive identity (often denoted as 0) and an identity with respect to multiplication is called a multiplicative identity (often denoted as 1). The distinction is used most often for sets that support both binary operations, such as rings. The multiplicative identity is often called the unit in the latter context, where, though, a unit is often used in a broader sense, to mean an element with a multiplicative inverse.

As the last example (a semigroup) shows, it is possible for (S, ∗) to have several left identities. In fact, every element can be a left identity. Similarly, there can be several right identities. But if there is both a right identity and a left identity, then they are equal and there is just a single two-sided identity. To see this, note that if l is a left identity and r is a right identity then l = l ∗ r = r. In particular, there can never be more than one two-sided identity. If there were two, e and f, then e ∗ f would have to be equal to both e and f.

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