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.

The term *identity element* is often shortened to *identity* when there is no possibility of confusion, and we will do so in this article.

Let `S` be a set 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**.

For example, if (`S`,*) denotes the real numbers with addition, then 0 is an identity. If (`S`,*) denotes the real numbers with multiplication, then 1 is an identity. If (`S`,*) denotes the `n`-by-`n` square matrices with addition, then the zero matrix is an identity. If (`S`,*) denotes the `n`-by-`n` matrices with multiplication, then the identity matrix is an identity. If (`S`,*) denotes the set of all functions from a set `M` to itself, with function composition as operation, then the identity map is an identity. If `S` has only two elements, `e` and `f`, and the operation * is defined by `e` * `e` = `f` * `e` = `e` and `f` * `f` = `e` * `f` = `f`, then both `e` and `f` are left identities, but there is no right or two-sided identity.

As the last example 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 `e` is an identity of (`S`,*) and `a` * `b` = `e`, then `a` is called a **left inverse** of `b` and `b` is called a **right inverse** of `a`. If an element `x` is both a left inverse and a right inverse of `y`, then `x` is called a **two-sided inverse**, or simply an **inverse**, of `y`.

As with identities, it is possible for an element `y` to have several left inverses or several right inverses. `y` can even have several left inverses *and* several right inverses. However if the operation * is associative, then if `y` has both a left inverse and a right inverse, then they are equal.

See also: Additive inverse, Group, Monoid, Quasigroup.