Ring of Integers Modulo Composite is not Integral Domain

From ProofWiki
Jump to navigation Jump to search

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.

$\blacksquare$


Sources