Multiplicative Inverse in Ring of Integers Modulo m/Proof 2

From ProofWiki
Jump to navigation Jump to search

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.

$\blacksquare$