Modulo Addition has Identity
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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 $m$.
$\blacksquare$
Sources
- 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