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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.

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  • $\begingroup$ 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 $\endgroup$ – user317953 Feb 26 '16 at 7:23
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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 $\bigoplus\limits_{i} Q/R$.

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