# Ring of Integers Modulo Composite is not Integral Domain

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## Theorem

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

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

Let $m$ be a composite number.

Then $\struct {\Z_m, +, \times}$ 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:

 $\displaystyle \eqclass 0 m$ $=$ $\displaystyle \eqclass m m$ $\displaystyle$ $=$ $\displaystyle \eqclass {k l} m$ $\displaystyle$ $=$ $\displaystyle \eqclass k m \times \\eqclass l m$

So $\struct {\Z_m, +, \times}$‎ is a ring with zero divisors.

So by definition $\struct {\Z_m, +, \times}$‎ is not an integral domain.

$\blacksquare$