Prove that there is a natural bijection between the set of $R[x]$-module structures on $M$ and $\mathrm{End}_{R-\mathsf{Mod}}(M)$.

Question

From Algebra Chapter $0$ by Aluffi:

Suppose $\alpha: R \to \mathrm{End}_{\mathsf{Ab}}(M)$ is an $R$-module structure on $M$. I would like to extend this to $R[x] \to \mathrm{End}_{\mathsf{Ab}}(M)$ which would be an $R[x]$-module structure on $M$.

This really amounts to defining how the polynomial $x$ should act on an element $m \in M$, which I am not sure how to define.

How can this be done?

Answer 1

You are asking where multiplication by $x$ sends a general element $m \in M$, but have no place to send it. What if I gave you a map $f : M \to M$? Any guesses on where it should go?