Integers Modulo m under Multiplication form Commutative Monoid
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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): $\S 1.1$