Complexification of $\mathbb{Z}$ using tensor products

Question

I would just like a brief explanation of how to evaluate, the following tensor product when the vector space $V$ is replaced by the set of integers $\mathbb{Z}$.

$$ ? = V\otimes_{\mathbb{R}} \mathbb{C} $$

Answer 1

Let $R$ be a ring. Given a right $R$-module $M$ and a left $R$-module $N$, we can form their tensor product denoted $M\otimes_R N$. If $R$ is a field, then a module over $R$ is simply a vector space over that field.

As $\mathbb{Z}$ is not a vector space over $\mathbb{R}$, it is not an $\mathbb{R}$-module, so the expression $\mathbb{Z}\otimes_{\mathbb{R}}\mathbb{C}$ is meaningless.