identity element
Identity element
An identity element of a binary operation is an element that leaves unchanged ever element when the operation is applied.
Definition
Consider \[\mathbb{Z}\]. It is common knowledge that \[0+a=a=a+0\] and \[1\times a=a=a\times 1\].
Let \[\ast\] be a binary operation on set \[S\]. We say that \[e\in S\] is an identity element for \[S\] if \[\forall\,a\in S,\,e\ast a=a=a\ast e\].
Thus, we can also show that if \[e,f\in S\] are identity elements for \[S\] with respect to \[\ast\], it follows that \[e=f\]. Since \[e\] and \[f\] are identity elements \[e\ast f=f\] and \[e\ast f=e\]. Hence, \[e=e\ast f=f\].