Ring of Integers Modulo m is Ring

Theorem
For all $m \in \N: m \ge 2$, the algebraic structure $\left({\Z_m, +_m, \times_m}\right)$ is a commutative ring with unity $\left[\!\left[{1}\right]\!\right]_m$.

The zero of $\left({\Z_m, +_m, \times_m}\right)$ is $\left[\!\left[{0}\right]\!\right]_m$.

Proof
First we check the ring axioms:


 * $A$: The Additive Group of Integers Modulo m $\left({\Z_m, +_m}\right)$ is an abelian group.


 * From Modulo Addition has Identity, $\left[\!\left[{0}\right]\!\right]_m$ is the identity of the additive group $\left({\Z_m, +_m}\right)$.

From Integers Modulo m under Multiplication form Commutative Monoid:


 * $M0$: $\left({\Z_m, \times_m}\right)$ is closed.
 * $M1$: $\left({\Z_m, \times_m}\right)$ is associative.
 * $M2$: $\left({\Z_m, \times_m}\right)$ has an identity $\left[\!\left[{1}\right]\!\right]_m$.
 * $C$: $\left({\Z_m, \times_m}\right)$ is commutative.

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

Also see

 * Canonical Epimorphism from Integers by Principal Ideal