Product Inverse in Ring is Unique

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

Let $x \in R$ be a unit of $R$.

Then the product inverse $x^{-1}$ of $x$ is unique.

Proof
By definition of ring with unity, the algebraic structure $\left({R, \circ}\right)$ is a monoid.

The result follows from Inverses in Monoid are Unique.