Definition:Modulo Addition/Definition 1

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

Let $\Z_m$ be the set of integers modulo $m$:
 * $\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$

where $\eqclass x m$ is the residue class of $x$ modulo $m$.

The operation of addition modulo $m$ is defined on $\Z_m$ as:
 * $\eqclass a m +_m \eqclass b m = \eqclass {a + b} m$

Also denoted as
Although the operation of addition modulo $m$ is denoted by the symbol $+_m$, if there is no danger of confusion, the symbol $+$ is often used instead.

The notation for addition of two integers modulo $m$ is not usually $\eqclass a m +_m \eqclass b m$.

What is more normally seen is $a + b \pmod m$.

Also see

 * Equivalence of Definitions of Integer Modulo Addition