Definition:Identity (Abstract Algebra)

Let $$\left({S, \circ}\right)$$ be an algebraic structure.

Left Identity
An element $$e_L \in S$$ is called a left identity iff:

$$\forall x \in S: e_L \circ x = x$$

Right Identity
An element $$e_R \in S$$ is called a right identity iff:

$$\forall x \in S: x \circ e_R = x$$

Identity
An element $$e \in S$$ is called a two-sided identity or simply identity iff it is both a left identity and a right identity.

Comment
Some authors use the term neutral element for identity. Others use unity but that's just too easy to confuse with other usages of this term. Similarly, some treatments refer to this as zero in certain circumstances, but it is best to reserve that term for when it is particularly appropriate.