Tensor product of $\mathbb{R}^d$ and $\mathbb{R}^s$ as abelian groups

Question

It is well known (and easy to prove) that $\mathbb{R}^d\otimes_{\mathbb{R}} \mathbb{R}^s$ is isomorphic as a vector space to $\mathbb{R}^{sd}$. Now, I would like to know a simple description of the tensor product of $\mathbb{R}^d$ and $\mathbb{R}^s$ as abelian groups, i.e., what is $\mathbb{R}^d\otimes_{\mathbb{Z}} \mathbb{R}^s$?

Answer 1

It is enough to describe $\mathbb{R} \otimes_{\mathbb{Z}} \mathbb{R}$. If $c=|\mathbb{R}|$, there is an isomorphism of $\mathbb{Q}$-vector spaces, hence of abelian groups $\mathbb{R} \cong \mathbb{Q}^{(c)}$. It follows $\mathbb{R} \otimes_{\mathbb{Z}} \mathbb{R} \cong \mathbb{Q}^{(c \times c)} \cong \mathbb{Q}^{(c)} \cong \mathbb{R}$. Notice that this is just a highly non-canonical isomorphism, probably of no use at all. When you want to work with $\mathbb{R} \otimes_{\mathbb{Z}} \mathbb{R}$, better leave it as a tensor product.