Definition:Integers Modulo m

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

The quotient set of congruence modulo $m$ is:


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

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 Integer is Congruent to 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.

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

It is usual, when it is unlikely that confusion will result, to denote the residue classes just by their defining integers:
 * $\Z_m = \left\{{0, 1, 2, \ldots, m-1}\right\}$

Also see

 * Definition:Congruence (Number Theory)/Integers


 * Definition:Modulo Addition
 * Definition:Modulo Multiplication