Unity of Ring is Unique

Theorem
A ring can have no more than one unity.

Proof
Let $\left({R, +, \circ}\right)$ be a ring.

If $\left({R, \circ}\right)$ has an identity, then it is a monoid.

From Identity of Monoid is Unique, it follows that such an identity is unique.

Also see

 * Identity is Unique