Ring of Integers Modulo m is Ring

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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:

$A$: 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:

$M0$: $\struct {\Z_m, \times_m}$ is closed.
$M1$: $\struct {\Z_m, \times_m}$ is associative.
$M2$: $\struct {\Z_m, \times_m}$ has an identity $\eqclass 1 m$.
$C$: $\struct {\Z_m, \times_m}$ is commutative.

Then:

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

$\blacksquare$


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Sources