# Characteristic of Ring with No Zero Divisors/Proof 1

## 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

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

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 24$