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Free modules over a ring generalise to projective modules over a ring, which generalise to flat modules, which generalise to torsion free modules: $$ \textrm{free} \to \textrm{projective} \to \textrm{flat} \to \textrm{torsion free}. $$ Does there exist an analogous chain of properties for injective modules.

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    $\begingroup$ There is a notion of a cofree module and there's also a notion of a divisible module over any type of ring, in the sense that appears in T.Y. Lam's Lectures on modules and rings around p 70. The last one always seemed like a fair dual to flatness. I've forgotten a dual for torsion-free modules, if I knew one. So at least Cofree -> injective -> divisible. $\endgroup$
    – rschwieb
    Apr 1, 2020 at 17:29

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