Is a divisible module over a local principal ideal domain a torsion module?

Question

Is an injective module over a local principal ideal domain a torsion module? We know that injective modules and divisible modules over a PID are equivalent. What do we say about the torsion submodule of such a module, in general? Thank you.

Answer 1

Injective modules over noetherian rings are classified, this was done by Matlis in his famous paper. Over a local PID $R$ the classification gets very easy, any injective module is of the form $\bigoplus\limits_{j} Q \oplus \bigoplus\limits_{i} Q/R$, where $Q$ is the fraction field of $R$.

The torsion submodule is of course equal to $\bigoplus\limits_{i} Q/R$.