Consider the $\mathbb{Z}$-modules $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}^n,\mathbb{Z})$ and $\mathbb{R}$. I want to understand the tensor product $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}^n,\mathbb{Z}) \otimes_{\mathbb{Z}} \mathbb{R}$. I was hoping to write it in the form $\operatorname{Hom}_{\mathbb{Z}}(\cdot,\cdot)$. Ideally, the equality $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}^n,\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{R}=\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}^n,\mathbb{R})$ would help us, but I do not think that this equality holds (as it generally does not hold for infinite dimensional spaces).
Any suggestions on how to think of this tensor product? Is it possible to express it as something of the form $\operatorname{Hom}_{\mathbb{Z}}(\cdot,\cdot)$?
Thank you in advance.