Definition:Additive Group of Integers Modulo m

Definition
Let $m \in \Z$ such that $m > 1$.

The additive group of integers modulo $m$ $\left({\Z_m, +_m}\right)$ is the set of integers modulo $m$ under the operation of addition modulo $m$.

Also see

 * Integers Modulo m under Addition form Cyclic Group

Thus addition modulo $m$ is:


 * Well-defined on $\Z_m$
 * Closed on $\Z_m$
 * Associative on $\Z_m$
 * Commutative on $\Z_m$
 * The identity of $\left({\Z_m, +_m}\right)$ is $\left[\!\left[{0}\right]\!\right]_m$.
 * The inverse of $\left[\!\left[{k}\right]\!\right]_m \in \Z_m$ is $\left[\!\left[{n - k}\right]\!\right]_m$.