Unity of Integral Domain is Unique

Theorem

Let $\struct {D, +, \times}$ be an integral domain.

Then the unity of $\struct {D, +, \times}$ is unique.

Proof

From the definition of an integral domain, $\struct {D, +, \times}$ is a commutative ring such that $\struct {D^*, \circ}$ is a monoid.

The result follows from Identity of Monoid is Unique.

$\blacksquare$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 4$. Elementary Properties: Theorem $1 \ \text{(ii)}$