Ring of Integers Modulo m is Ring

Theorem
For all $m \in \N: m \ge 2$, the ring of integers modulo $m$:
 * $\struct {\Z_m, +_m, \times_m}$

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:


 * 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 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:


 * $\struct {\Z_m, \times_m}$ is closed.
 * $\struct {\Z_m, \times_m}$ is closed.


 * $\struct {\Z_m, \times_m}$ is associative.
 * $\struct {\Z_m, \times_m}$ is associative.


 * $\struct {\Z_m, \times_m}$ has an identity $\eqclass 1 m$.
 * $\struct {\Z_m, \times_m}$ has an identity $\eqclass 1 m$.


 * $\struct {\Z_m, \times_m}$ is commutative.
 * $\struct {\Z_m, \times_m}$ is commutative.

Then:
 * $\times_m$ distributes over $+_m$ in $\Z_m$.
 * $\times_m$ distributes over $+_m$ in $\Z_m$.

Also see

 * Canonical Epimorphism from Integers by Principal Ideal