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) $$p$$ must be a prime number;
 * 2) $$p$$ 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.