Product Inverse in Ring is Unique

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.

$\blacksquare$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Fields: $\S 14$. Definition of a Field