Modulo Addition has Identity

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:

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

$\ds \eqclass x m +_m \eqclass 0 m$

$=$

$\ds \eqclass {x + 0} m$

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 $m$.

$\blacksquare$

1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: A Little Number Theory
1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): $\S 1$: Some examples of groups: Example $1.10$
1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 6$. The Residue Classes: Theorem $5$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 19.1$: Properties of $\Z_m$ as an algebraic system