A module $M$ over a commutative ring $R$ is called a 'injective module' if it satisfies certain universal property explaned here.

Question: Is there any intuition how to think concretely about injective modules? Do them naturally arise as an attempt go generalize a special class of modules? I'm asking this because I try to find an analogy to the dual concept of projective modules.

Although these are formally defined by a similar (but dual) UP these have a more accessible interpretation: These arise as a natural generalization of free modules and form literally finer building blocks of free modules since there is a fact that a module is projective iff it is a direct summand of a free module.

Does these exist a similar interpretation for injective modules? Which class of modules do these naturally generalize and do they arise also as 'building blocks' of something?

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    $\begingroup$ Injective modules are the dual of projective modules. In the case of $\mathbb{Z}$-modules, they are precisely the divisible abelian groups. $\endgroup$ Feb 25 at 1:30
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    $\begingroup$ That's not a universal property btw. $\endgroup$
    – Qi Zhu
    Feb 25 at 5:51

I've always thought of injective modules this way (beyond simply a dual definition to projective modules):

  1. Injective modules are a summand of any module containing them
  2. Every module embeds in an injective module, which is a kind of "completion"

Now, I have not worked with cofree modules, but you may want to check out this entry on cofree modules. On one hand, I believed that such a dual description exists, on the other hand, I don't have personal experience with it, and I don't find wolfram mathworld to be very reliable. So please take it with a grain of salt.

It's also worth noting that the second bullet above does not dualize: the dual would usually be considered to be "every module has a projective cover" but it is not true. However, a famous result is that every module has a flat cover. Rings for which every right module has a projective cover are known as right perfect rings.

  • $\begingroup$ The cofree $R$-modules are presicely the products of copies of $Hom(R, Q/Z)$'s. Also, $Q, Q/Z$ are the first (and unfortunately the last) examples of injective modules i know. Are there more didacticaly important examples of injective modules one could have in mind? Since all injetive modules are direct summands of every module that includes it, the question is esentially which submodules do have products of the $Hom(R, Q/Z)$'s and how do they look like? Are there any more interesting then $Q$ and $ Q/Z$ ? $\endgroup$ Feb 27 at 1:14

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