Integers Modulo m under Multiplication form Commutative Monoid

Theorem

Let $\struct {\Z_m, \times_m}$ denote the algebraic structure such that:

$\Z_m$ is the set of integers modulo $m$

$\times_m$ denotes the operation of multiplication modulo $m$.

Then $\struct {\Z_m, \times_m}$ is a commutative monoid.

Proof

Multiplication modulo $m$ is closed.

Multiplication modulo $m$ is associative.

Multiplication modulo $m$ has an identity:

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

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): $\S 1.1$