proof that every subring of a cyclic ring is a cyclic ring

The following is a proof that every subring of a cyclic ring is a cyclic ring.

Proof.

Let $R$ be a cyclic ring and $S$ be a subring of $R$. Then the additive group of $S$ is a subgroup of the additive group of $R$. By definition of cyclic group, the additive group of $R$ is cyclic (http://planetmath.org/CyclicGroup). Thus, the additive group of $S$ is cyclic. It follows that $S$ is a cyclic ring.∎