Ring of Integers Modulo m is Ring

From ProofWiki
Jump to navigation Jump to search

Theorem

For all $m \in \N: m \ge 2$, the ring of integers modulo $m$:

$\struct {\Z_m, +_m, \times_m}$

is a commutative ring with unity $\eqclass 1 m$.


The zero of $\struct {\Z_m, +_m, \times_m}$ is $\eqclass 0 m$.


Proof

First we check the ring axioms:

$\text A$: The Integers Modulo $m$ under Addition form Abelian Group.
From Modulo Addition has Identity, $\eqclass 0 m$ is the identity of the additive group $\struct {\Z_m, +_m}$.

From Integers Modulo m under Multiplication form Commutative Monoid:

$\text M 0$: $\struct {\Z_m, \times_m}$ is closed.
$\text M $: $\struct {\Z_m, \times_m}$ is associative.
$\text M 2$: $\struct {\Z_m, \times_m}$ has an identity $\eqclass 1 m$.
$\text C$: $\struct {\Z_m, \times_m}$ is commutative.

Then:

$\text D$: $\times_m$ distributes over $+_m$ in $\Z_m$.

$\blacksquare$


Also see


Sources