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

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.

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

Also see

 * Canonical Epimorphism from Integers by Principal Ideal