# Ring of Integers Modulo m is Ring

## Theorem

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

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

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

## Proof

First we check the ring axioms:

Ring Axiom $\text A$: Addition forms an Abelian Group
The Integers Modulo $m$ under Addition form Abelian Group.
From Modulo Addition has Identity, $\eqclass 0 m$ is the identity of the additive group $\struct {\Z_m, +_m}$.
Ring Axiom $\text M0$: Closure under Product:
$\struct {\Z_m, \times_m}$ is closed.
Ring Axiom $\text M1$: Associativity of Product:
$\struct {\Z_m, \times_m}$ is associative.
Ring Axiom $\text M2$: Identity Element for Ring Product:
$\struct {\Z_m, \times_m}$ has an identity $\eqclass 1 m$.
Ring Axiom $\text C$: Commutativity of Ring Product:
$\struct {\Z_m, \times_m}$ is commutative.

Then:

Ring Axiom $\text D$: Distributivity of Product over Addition:
$\times_m$ distributes over $+_m$ in $\Z_m$.

$\blacksquare$