Ordered Integral Domain is Totally Ordered Ring

Theorem

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

Then $\struct {D, +, \times, \le}$ is a totally ordered ring.

Proof

By definition, $\struct {D, +, \times, \le}$ is an integral domain endowed with a strict positivity property.

From Strict Positivity Property induces Total Ordering, the ordering $\le$ on $\struct {D, +, \times, \le}$ is a total ordering.

Hence the result by definition of totally ordered ring.

$\blacksquare$