Integers Modulo m under Multiplication form Commutative Monoid

Theorem
The structure:
 * $\left({\Z_m, \times}\right)$

(where $\Z_m$ is the set of integers modulo $m$) is a commutative monoid.

Proof

 * Multiplication modulo $m$ is closed.


 * Multiplication modulo $m$ is associative.


 * Multiplication modulo $m$ has an identity:


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

This identity is unique.


 * Multiplication modulo $m$ is commutative.


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