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. It leaves other elements unchanged when combined with them. This is used for groups and related concepts.

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 (known as a magma). 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.

Read more about Identity Element: Examples, Properties

### Famous quotes containing the words identity and/or element:

“I look for the new Teacher that shall follow so far those shining laws that he shall see them come full circle; shall see their rounding complete grace; shall see the world to be the mirror of the soul; shall see the *identity* of the law of gravitation with purity of the heart; and shall show that the Ought, that Duty, is one thing with Science, with Beauty, and with Joy.”

—Ralph Waldo Emerson (1803–1882)

“The geometry of landscape and situation seems to create its own systems of time, the sense of a dynamic *element* which is cinematising the events of the canvas, translating a posture or ceremony into dynamic terms. The greatest movie of the 20th century is the Mona Lisa, just as the greatest novel is Gray’s Anatomy.”

—J.G. (James Graham)