Definition:Identity (Abstract Algebra)

Given a set $$S$$ together with a binary operation $$\circ$$, an element $$e \in S$$ is called a left identity if $$e \circ a=a$$ for each $$a \in S$$ and a right identity if $$b \circ e=b$$ for each $$b \in S$$.

If $$e$$ is both a right and left identity, it is called a two-sided identity or simply identity.