Modulo Multiplication has Identity

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Theorem

Multiplication modulo $m$ has an identity:

$\forall \eqclass x m \in \Z_m: \eqclass x m \times_m \eqclass 1 m = \eqclass x m = \eqclass 1 m \times_m \eqclass x m$


Proof

Follows directly from the definition of multiplication modulo $m$:

\(\ds \eqclass x m \times_m \eqclass 1 m\) \(=\) \(\ds \eqclass {x \times 1} m\)
\(\ds \) \(=\) \(\ds \eqclass x m\)
\(\ds \) \(=\) \(\ds \eqclass {1 \times x} m\)
\(\ds \) \(=\) \(\ds \eqclass 1 m \times_m \eqclass x m\)


Thus $\eqclass 1 m$ is the identity for multiplication modulo $m$.

$\blacksquare$


Sources

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