Identity for a binary operation (reviewed)

If $*$ is a binary operation on a set $A$, then $A$ has an identity element with respect to $*$ if there exists $e\in A$ such that for all $a\in A$, $$ a*e = e*a = a.$$ Such an identity element $e$, if it exists, is unique and is thus called the identity element of $A$ with respect to $*$.