# Injective Modules Motivation & Intuition

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?

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

• 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 $Q/Z$ ? – Isak the XI Feb 27 at 1:14