Characteristic of Finite Ring with No Zero Divisors

Theorem
Let $\left({R, +, \circ}\right)$ be a finite ring with unity with no (proper) zero divisors whose zero is $0_R$ and whose unity is $1_R$.

Let $n \ne 0$ be the characteristic of $R$.

Then:
 * $(1): \quad n$ must be a prime number
 * $(2): \quad n$ is the order of all non-zero elements in $\left({R, +}\right)$.

It follows that $\left({R, +}\right) \cong C_n$, where $C_n$ is the cyclic group of order $n$.

Proof 1
Follows directly from Subring Generated by Unity of Ring with Unity.

Proof 2
Suppose $\operatorname{Char} \left({R}\right) = n$ where $n$ is composite.

Let $n = r s$, where $r, s \in \Z, r > 1, s > 1$.

First note that:

Then:

... as $R$ has no (proper) zero divisors.

But both $r$ and $s$ are less than $n$ which contradicting the minimality of $n$.

So if $\operatorname{Char} \left({R}\right) = n$ it follows that $n$ must be prime.

Now let $x \in R^*$.

Then by Characteristic Times Element of Ring is Zero, $n \cdot x = 0_R$.

It follows from Element to the Power of Multiple of Order that $\left|{x}\right| \backslash n$.

Since $n$ is prime, $\left|{x}\right| = 1$ or $\left|{x}\right| = n$.

It can't be $1$, from Characteristic of Null Ring is One, so the result follows.