ring structure on free abelian groups

Question

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:

1. How many commutative ring structures is there on the group $\mathbb{Z}^{n}$ up to ring isomorphism.
2. 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.

Answer 1

At least countably many commutative structures when $n\geq 2$. Just take $\mathbb{Z}[x]/(f(x))$ where $f(x)\in \mathbb{Z}[x]$ is a monic irreducible polynomial of degree $n$. (There are countably many different extensions of $\mathbb{Q}$ of degree $n$, so while some of these rings are isomorphic, there are countably many different ones.)

On the other hand there are at most countably many non-commutative ring structures when $n\geq 2$. Each such ring structure is of the form $\mathbb{Z}\left\{x_1,x_2,\ldots, x_{n}\right\}/I$ where $I$ is the ideal generated by relations $x_i x_j = \sum_{k=1}^{n}a_{i,j,k} x_k$ with $a_{i,j,k}\in \mathbb{Z}$. There are countably many such choices for these relations.