Non-Zero Modulo Numbers Closed under Multiplication then Modulo is Prime

Theorem
Let $\struct {\Z_m, +_m, \times_m}$ be the ring of integers modulo $m$ for $m > 1$.

Let $\Z'_m$ be the set of non-zero integers modulo $m$.

Let $\struct {\Z_m, \times_m}$ be closed under modulo multiplication.

Then $m$ is prime.

Proof
Suppose $m$ is not prime.

Then $m = r s$ for some $r, s \in \Z: 1 < r < m, 1 < s < m$.

So $r, s \in \Z'_m$.

But:
 * $r \times_m s \equiv 0 \pmod m$

and so $r \times_m s \notin \Z'_m$.

So if $m$ is not prime, $\struct {\Z_m, \times_m}$ is not closed.

The result follows from the Rule of Transposition.