Multiplicative Inverse in Ring of Integers Modulo m/Proof 2

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.