identity element

The identity element of a binary operation is the element such that combining it with any other element under the operation returns simply the other element.

Formally,
let set $S$ be quipped with a binary operation $\ast$.
An element $e\in S$ is

• the left identity if $e\ast s=s$ for all $s\in S$,
• the right identity if $s\ast e=s$ for all $s\in S$,
  and a two-sided identity or simply identity when both are true.

The identity element of addition, $0$ ("zero"), is the additive identity.
The identity element of multiplication, $1$ ("unity" or "one"), is the multiplicative identity.

The identity element of a group is unique.