Characteristic of Finite Ring with No Zero Divisors

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

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

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

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

Proof

 * Follows directly from Subring Generated by Unity of Ring with Unity.


 * Alternatively, let $\operatorname{Char} \left({R}\right) = n = r s$, where $r, s \in \Z, r \ne 0, s \ne 0$.

First note that:

Then:

... so contradicting the minimality of $n$.


 * Let $x \in R^*$.

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.