# Product Inverse in Ring is Unique

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## Theorem

Let $\struct {R, +, \circ}$ 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 $\struct {R, \circ}$ is a monoid.

The result follows from Inverse in Monoid is Unique.

$\blacksquare$