# Generator of Additive Group Modulo m iff Unit of Ring

## Theorem

Let $m \in \Z: m > 1$.

Let $\struct {\Z_m, +_m}$ denote the additive group of integers modulo $m$.

Let $\struct {\Z_m, +_m, \times_m}$‎ be the ring of integers modulo $m$.

Let $a \in \Z_m$.

Then:

$a$ is a generator of $\struct {\Z_m, +_m}$
$a$ is a unit of $\struct {\Z_m, +_m, \times_m}$

## Proof

From Integers under Addition form Infinite Cyclic Group, the identity element $1_\Z$ of the ring of integers $\struct {\Z, +, \times}$ is a generator of the group $\struct {\Z, +}$.

Thus from Quotient Group of Cyclic Group, the identity element $1_{\Z_m}$ of the ring $\struct {\Z_m, +_m, \times_m}$ is a generator of the group $\struct {\Z_m, +_m}$.

Let $a \in \Z_m$.

Suppose $1_{\Z_m} \in \gen a$, where $\gen a$ signifies the group generated by $a$.

Then the smallest subgroup of $\struct {\Z_m, +_m}$ containing $1_{\Z_m}$, that is $\struct {\Z_m, +_m}$ itself, is contained in $\gen a$.

Thus $\gen a = \struct {\Z_m, +_m}$ iff $1_{\Z_m} \in \gen a$.

However, from Subgroup of Additive Group Modulo m is Ideal of Ring, $\gen a$ is an ideal of $\struct {\Z_m, +_m, \times_m}$, and hence is the principal ideal $\ideal a$ generated by $a$.

But from Principal Ideal from Element in Center of Ring, $1_{\Z_m} \in \gen a$ if and only if $a$ is a unit of the ring $\struct {\Z_m, +_m, \times_m}$.

Hence the result.

$\blacksquare$