Integers Modulo m under Multiplication form Commutative Monoid

Theorem

The structure:

$\struct {\Z_m, \times}$

(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:

$\forall k \in \Z: \eqclass k m \eqclass 1 m = \eqclass k m = \eqclass 1 m \eqclass k m$

This identity is unique.

Multiplication modulo $m$ is commutative.

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

$\blacksquare$

Sources

• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\S 1.1$