I'm finding finite rings $(R, +, \times$) such that $(R,+)$ is cyclic. $\mathbb{Z}_n$ is a good example. Up to ring isomorphism, is there any exmaples other than $\mathbb{Z}_n$ ?
Thanks.
For any finite cyclic group $(G,+)$, there exists $n$ such that $(\mathbb{Z}_n, +)$ is isomorphic to $(G,+)$.
Edit: Thanks to those commenting this post. It disappoints me that the answer is NO.
Simply to sum up the comments and answer the case where $R$ doesn't have a unit.
If $R$ has a unit, then $1$ must be a generator of $(R,+)$. Indeed, let $k$ be the smallest integer such that $k\cdot 1 = 0$. Then for all $g\in R$, $k\cdot g =0$, so that $|R| = k$, and thus since $\{0,...,k-1\} \to R$, $j\to j\cdot 1$ is injective (by minimality of $k$) it is also surjective (by equality of cardinals).
This way, $\phi : \mathbb{Z}/k\mathbb{Z} \to R$ defined by $\phi(j) = j\cdot 1$ is well defined and a group isomorphism.
Moreover, $\phi(ab) = \phi(a\cdot b) = a\cdot \phi(b) = (a\cdot 1)\phi(b) = \phi(a)\phi(b)$ by distributivity, so $\phi$ is a ring isomorphism, which gives us the desired result.
However this does not hold if $R$ has no unit. Indeed there are finite cyclic rings with no unit, and they obviously cannot be isomorphic to rings with a unit (for instance, take $2\mathbb{Z}/4\mathbb{Z}$ which is a ring (since it's an ideal) without a unit, since $2\times 2 =0$, and it's obviously cyclic (with generator $2$)). A question would be : what are the finite unitless rings with cyclic additive group ?
To answer this, let $(R,+,×)$ be such a ring, and let $a$ be a generator of $(R,+)$. $a^2 = p\cdot a $ for some $p\in \mathbb{N}$ (choose $p$ to be the smallest such for instance). Then take $m = p|R|$. We want to show that $R\simeq p\mathbb{Z}/m\mathbb{Z}$. Consider $\phi : R\to p\mathbb{Z}/m\mathbb{Z}$ the group isomorphism given by $\phi(a) = p$( well defined because $a$ generates $R$ and $p$ generates $p\mathbb{Z}/m\mathbb{Z}$ and they have the same order).
Then $\phi(cd)= \phi((k\cdot a)(l\cdot a)) = \phi(kl\cdot a^2) = \phi(pkl\cdot a) = pkl p= pk pl = \phi(a)\phi(b)$, so that $\phi$ is actually a ring isomorphism.
Therefore the complete classifiction of finite rings with cyclic additive groups is : any such ring is isomorphic to one $p\mathbb{Z}/m\mathbb{Z}$ with $p\mid m$. I haven't thought about the unicity of this representation (it is obviously unique in the unitary case, but what about the second case?)