Characteristic_of_Finite_Ring_with_No_Zero_Divisors

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 $n$.