Let $R$ be a PID. Prove that restriction-of-scalars induces an equivalence of categories of finitely generated modules $$\coprod_p (R_{(p)}-\mathrm{mod})_{tors}\to (R-\mathrm{mod})_{tors},$$ where the coproduct is taken over the primes in $R$ up to association.
I have no idea how to approach this question. I know that if $\varphi_p: R \to (R - p)^{-1}R$ is the restriction of scalars and $M$ is an $R$-module, then $(R - p)^{-1}R \otimes_R M \cong (R-p)^{-1}M$. However, I am not sure where $R$ being a PID comes into play, as well as how to how the equivalence of categories. I know that a functor $F$ is an quivalence of categories iff it's fully faithful and essentially surjective. I know that any finitely generated modules over the PID $R$ is isomorphic to $R/(p_1) \times \cdots, R/(p_n)$ for some $n \in \mathbb{N}$. I'm guessing that the "tors" subscribtion means that all components in the product is torsion, i.e. in this case equivalent to finitely generated.