This is a problem from my lecture notes, and I don't really know how to tackle it. Any help will be appreciated.
The question reads as follow:
Problem
Given an integral domain $R$, and an $R$-module $M$. Is the following statement true:
If $\mbox{Ext}(R/aR; M) = 0, \forall a \in R$, then $M$ is divisible.
In fact, the problem has 2 parts, the first part asks to prove:
If $M$ is divisible, then $\mbox{Ext}(R/aR; M) = 0, \forall a \in R$, which I've finally managed to do it.
Back to the second part, I think the statement is true. And what I'm currently struggling is that I don't really know how to relate $M$, and $R$. I've tried is to find a special module $W_m$, for each $m \in M$, such that, when knowing this exact sequence splits $0 \rightarrow M \xrightarrow{\chi} W_m \xrightarrow{\sigma} R/aR \rightarrow 0$, I can prove that $m$ is divisible by $a$. But as I don't know how to relate $M$, and $R$, I don't think that's the correct way to do it.
Thanks you guys a lot,
And have a good day,