Multiplicative Inverse in Ring of Integers Modulo m/Proof 2

Theorem
Let $\left({\Z_m, +_m, \times_m}\right)$ be the Ring of Integers Modulo m.

Then $\left[\!\left[{k}\right]\!\right]_m \in \Z_m$ has an inverse in $\left({\Z_m, \times_m}\right)$ iff $k \perp m$.

Proof
We have that the Ring of Integers Modulo m 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.