Classifying torsion-free injective modules over a PID

Question

So the question is as in the title:

Let $R$ be a PID. Classify all torsion-free injective modules.

I know that it is going to be divisible, and using torsion-free, if we define $\varphi_r:M\rightarrow M$ via $\varphi_r(m)=rm$ then we get that $\varphi_r$ is an isomorphism for all $r$ non-zero. That is, $rM\cong M$, but I don't know what else can I say about it.

Answer 1

So given $M$ an injective torsion-free module, we have that $\varphi_r$ is an isomorphism. Note that we can view $M$ as a $K(R)$-module (the field of fractions) in the following manner:

Define $(a/b)\cdot m$ to be $\varphi^{-1}_b(am)$, as $\varphi_b$ is an isomorphism this is well defined. Hence, we have that $M$ is a $K(R)$-module, that is, a $K(R)$-vector space.

Hence, the injective torsion-free modules are precisely vector spaces over $K(R)$.