Let $\mathbb{Z}^{n}$ be the free abelian group of rank $n$.
A ring structure on $\mathbb{Z}^{n}$ is a choice of a unit element $e\in \mathbb{Z}^{n} $ and a bilinear map $m:\mathbb{Z}^{n}\otimes_{\mathbb{Z}}\mathbb{Z}^{n}\rightarrow \mathbb{Z}^{n}$ satisfying the standard axiomes for ring structure.
My question is the following:
- How many commutative ring structures is there on the group $\mathbb{Z}^{n}$ up to ring isomorphism.
- How many ring structures is there on the group $\mathbb{Z}^{n}$ up to ring isomorphism.
I feel that the answer should depend on $n$ in essential way, but I am not sure. I will be happy for any classification result in low dimension.