# Classify all rings with cyclic additive group

Classify all rings (with unit or not) whose additive group is cyclic.

Let $g$ be the generator of the additive group. Then we have a surjective group homomorphism $\phi$ from $\mathbb Z$ to $A$ which sends $1$ in $g$. So $A$ is isomorphic, as group, to some $\mathbb Z/n\mathbb Z$, $n\in \mathbb Z$ with isomorphism $\bar\phi$. One also gets (by applying the distributive property) that $\phi(n)\phi(m)=\phi(nm) g^2$.

What can be said now? Non-isomorphic rings are those where the additive order of $g$ is different. But also if that order is the same, I think there can be non-isomorphic rings. My claim is that the multiplicative order (in the ring $\mathbb Z/n\mathbb Z$) of $\bar\phi^{-1}(g^2)$ classifies these rings. Is this true?

And if not, can someone give me a clue on how to deal with this very interesting problem?

In any case, the answer is that any $n$-element ring with cyclic additive group is isomorphic to $d\mathbb Z_{dn}$ for some divisor $d$ of $n$, and different divisors yield nonisomorphic rings. (When $d=n$ you get a zero ring.) An infinite ring with cyclic additive group is a zero ring or is isomorphic to $d\mathbb Z$ for some positive $d$, and different choices of $d$ yield nonisomorphic rings.