TITLE: Can we make the real numbers a $\mathbb{C}$-module?
QUESTION [5 upvotes]: Can we make the real numbers into a $\mathbb{C}$-module with unity? 

We should take the real numbers as a group and the complex numbers as our ring. Please tell me what is the operation?

REPLY [2 votes]: As an abelian group, $\mathbb{R}$ is uniquely divisible (that is, for each $x$ and nonzero integer $n$, there is a unique $y$ so that $x = n \cdot y$), so $\mathbb{R}$ is isomorphic to a vector space over $\mathbb{Q}$. (there's a more obvious proof that $\mathbb{R}$ is a rational vector space -- the point is that "uniquely divisible abelian group" means the same thing as "rational vector space", so I will talk about the latter instead)
In fact, (I'm surely assuming axiom of choice), $\mathbb{R}$ is a $\mathfrak{c}$-dimensional rational vector space, where $\mathfrak{c}$ is the cardinality of $\mathbb{R}$.
$\mathbb{C}$ is also a $\mathfrak{c}$-dimensional rational vector space. Therefore, $\mathbb{C} \cong \mathbb{R}$ as rational vector spaces, and thus as abelian groups.
Thus, your question is equivalent to the follwing:


Can we make the complex numbers into a $\mathbb{C}$-module with unity? 

We should take the complex numbers as a group and the complex numbers as our ring. Please tell me what is the operation?