Definition:Ring of Integers Modulo m

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 $\left[\!\left[{1}\right]\!\right]_m$.

Proof
First we check the ring axioms:


 * A: The Additive Group of Integers Modulo m $\left({\Z_m, +_m}\right)$ is a group.
 * M0: The Multiplicative Monoid of Integers Modulo m $\left({\Z_m, \times_m}\right)$ is closed.
 * M1: The Multiplicative Monoid of Integers Modulo m $\left({\Z_m, \times_m}\right)$ is associative.
 * D: $\times_m$ distributes over $+_m$ in $\Z_m$.

Then we note that Multiplicative Monoid of Integers Modulo m $\left({\Z_m, \times_m}\right)$ is commutative.

Then we note that the Multiplicative Monoid of Integers Modulo m $\left({\Z_m, \times_m}\right)$ has an identity $\left[\!\left[{1}\right]\!\right]_m$.

Finally we note that $\left[\!\left[{0}\right]\!\right]_m$ is the identity of the additive group $\left({\Z_m, +_m}\right)$.

Also see

 * Canonical Epimorphism from Integers by Principal Ideal