Characteristic of Finite Ring with No Zero Divisors

Theorem
Let $\struct {R, +, \circ}$ 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 $\struct {R, +}$.

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