Multiplicative Inverse in Ring of Integers Modulo m/Proof 2
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Theorem
Let $\struct {\Z_m, +_m, \times_m}$ be the ring of integers modulo $m$.
Then $\eqclass k m \in \Z_m$ has an inverse in $\struct {\Z_m, \times_m}$ if and only if $k \perp m$.
Proof
From Ring of Integers Modulo m is Ring, $\left({\Z_m, +_m, \times_m}\right)$ is a commutative ring with unity $\left[\!\left[{1}\right]\!\right]_m$.
Thus by definition $\left({\Z_m, \times_m}\right)$ is a commutative monoid.
The result follows from Multiplicative Inverse in Monoid of Integers Modulo m.
$\blacksquare$