a finite ring is cyclic if and only its order and characteristic are equal

. A finite ring is cyclic if and only if its order (http://planetmath.org/OrderRing) and characteristic are equal.

Proof.

If $R$ is a cyclic ring and $r$ is a generator (http://planetmath.org/Generator) of the additive group of $R$, then $|r|=|R|$. Since, for every $s\in R$, $|s|$ divides $|R|$, then it follows that $\mathrm{char}R=|R|$. Conversely, if $R$ is a finite ring such that $\mathrm{char}R=|R|$, then the exponent of the additive group of $R$ is also equal to $|R|$. Thus, there exists $t\in R$ such that $|t|=|R|$. Since $⟨t⟩$ is a subgroup of the additive group of $R$ and $|⟨t⟩|=|t|=|R|$, it follows that $R$ is a cyclic ring.∎