Ring of Integers Modulo Composite is not Integral Domain

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

Let $\left({\Z_m, +, \times}\right)$‎ be the ring of integers modulo $m$.

Let $m$ be a composite number.

Then $\left({\Z_m, +, \times}\right)$ is not an integral domain.

Proof
Now suppose $m \in \Z: m \ge 2$ be composite.

Then:
 * $\exists k, l \in \N_{> 0}: 1 < k < m, 1 < l < m: m = k \times l$

Thus:

So $\left({\Z_m, +, \times}\right)$‎ is a ring with zero divisors.

So by definition $\left({\Z_m, +, \times}\right)$‎ is not an integral domain.