Ring of Integers Modulo m cannot be Ordered Integral Domain

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Let $m \in \Z: m \ge 2$.

Let $\struct {\Z_m, +, \times}$‎ be the ring of integers modulo $m$.

Then $\struct {\Z_m, +, \times}$ cannot be an ordered integral domain.


First note that from Ring of Integers Modulo Prime is Integral Domain, $\struct {\Z_m, +, \times}$‎ is an integral domain only when $m$ is prime.

So for $m$ composite the result holds.

If $m$ is prime, and $\struct {\Z_m, +, \times}$ is therefore an integral domain, its order is finite.

The result follows from Finite Integral Domain cannot be Ordered.