I would like to know if there is some analogue theorem of structure for divisible modules as there is for divisible abelian groups.
More exactly, given a divisible $R$-module $M$, where $R$ is a PID, we know we can express $M$ as $M=t(M)\oplus N$, where $N\cong\frac{M}{t(M)}$. Since $N$ is torsion-free and also divisible, we can view $N$ as a vectorial space over $Q_{R}$ (here $Q_{R}$ is the field of fractions of $R$). So we only have to focus on $T(M)$. A basic result asserts that $T(M)$ is a direct sum of its $p$-primary components (here $p$ is an irreducible (or prime) element of $R$). My problem is, what about these $p$-primary components? These are not necessarily finitely generated, so is there any expression of them as a direct sum of some kind of $R$-modules?


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