Integers Modulo m under Multiplication form Commutative Monoid

Theorem
The structure $$\left({\mathbb{Z}_m, \times}\right)$$ is a commutative monoid.

Proof

 * Multiplication modulo $m$ is closed.


 * Multiplication modulo $m$ is associative.


 * Multiplication modulo $m$ has an identity:

$$\forall k \in \mathbb{Z}: \left[\!\left[{k}\right]\!\right]_m \left[\!\left[{1}\right]\!\right]_m = \left[\!\left[{k}\right]\!\right]_m = \left[\!\left[{1}\right]\!\right]_m \left[\!\left[{k}\right]\!\right]_m$$

This Identity is Unique.


 * Multiplication modulo $m$ is commutative.


 * Thus all the conditions are fulfilled for this to be a commutative monoid.