Definition:Integers Modulo m

Definition
Let $$m \in \Z$$ be an integer.

The quotient set of congruence modulo $m$ is:


 * $$\Z_m = \frac {\Z} {\mathcal R_m} = \left\{{\left[\!\left[{0}\right]\!\right]_m, \left[\!\left[{1}\right]\!\right]_m, \ldots, \left[\!\left[{m-1}\right]\!\right]_m}\right\}$$

where:


 * $$\mathcal R_m$$ is the equivalence relation defined as congruence modulo $m$;


 * $$\left[\!\left[{x}\right]\!\right]_m$$ is the residue class of $x$ modulo $m$.

Thus there are $$m$$ different residue classes modulo $m$.

From Congruence to an Integer less than Modulus, it follows that the set defined here is a complete repetition-free list of them.

This definition is a refinement of the concept of the set of all residue classes in the domain of real numbers.

This structure can also be rendered $$\left({\N_m, +_m}\right)$$, using $$\N_m$$ as defined in Subset of Natural Numbers.