# Opinions on **Identity element**

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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.

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Thanks to this graph, we can see the interest Identity element has and the evolution of its popularity.

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