# Ordered Integral Domain is Totally Ordered Ring

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## 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.

$\blacksquare$