Let $V$ be an $n$-dimensional complex vector space. In particular, it is an Abelian group. Let $R$ be a (commutative, unitary) $\mathbb C$-algebra.
Problem. I would like to parameterize all $R$-module structures one can put on $V$.
These are exactly, by definition, the elements of the set $$\hom_{\textrm{Ring}}(R,\textrm{End}_{\mathbb Z}(V)).$$ Unfortunately, I cannot do anything better. I do not even know if I can replace $\textrm{End}_{\mathbb Z}(V)$ by $\textrm{End}_{\mathbb C}(V)$, the ring of $\mathbb C$-linear endormorphisms of $V$, finally viewed as a vector space. Ideally, I would like to say that to give an $R$-module structure on $V$ is to give a tuple of endomorphisms of $V$, one for each generator of the algebra $R$.
Does anyone have any suggestion? Does the above problem have a well-known solution?
Thank you!