Multiplicative Monoid of Integers Modulo m

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.

$\blacksquare$