## Theorem

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

Then addition modulo $m$ has an identity:

$\forall \eqclass x m \in \Z_m: \eqclass x m +_m \eqclass 0 m = \eqclass x m = \eqclass 0 m +_m \eqclass x m$

That is:

$\forall a \in \Z: a + 0 \equiv a \equiv 0 + a \pmod m$

## Proof

 $\ds \eqclass x m +_m \eqclass 0 m$ $=$ $\ds \eqclass {x + 0} m$ Definition of Modulo Addition $\ds$ $=$ $\ds \eqclass x m$ $\ds$ $=$ $\ds \eqclass {0 + x} m$ $\ds$ $=$ $\ds \eqclass 0 m +_m \eqclass x m$ Definition of Modulo Addition

Thus $\eqclass 0 m$ is the identity for addition modulo $m$.

$\blacksquare$