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In almost all textbooks on Homological Algebra, when they talk about injective modules, they do not give many examples, usually are $\mathbb{Q}/\mathbb{Z}$ and $\mathbb{R}/\mathbb{Z}$. Is there a general way to construct an injective module (except using Baer criterion)?

The other question is, if a ring $R$ is hereditary then every submodule of a left $R$-projective module $M$ is projective. Then every quotient module of a left injective $R$-module is injective too. What about submodule of a left, injective $R$-module? Is it also injective? Please help me to find a counterexample (if it exists). Thanks.

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3 Answers 3

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I think you will have more luck learning nontrivial facts about and examples of injective modules in (non/)commutative algebra texts rather than homological algebra texts. In homological algebra texts, injective modules (and more generally injective objects in an abelian category) play an important technical role but are not really studied for their own sake. (Several times on this site and MO I have quipped that although I know how to prove that the category of sheaves on a topological space has enough injectives, nevertheless I have never met a nontrivial injective sheaf.)

My own lecture notes / proto-text on commutative algebra has a decently substantial section -- $\S 3.6$ -- on injective modules. If you need to know a lot about injective modules you will need to look elsewhere, but it sounds like you are frustrated by the fact that you have been told almost nothing about them, and for that my notes can serve as a remedy. In particular I discuss:

1) Suppose $R$ is a domain and $M$ is an $R$-module. Then:
a) If $M$ is injective, then $M$ is divisible. In particular, $\mathbb{Z}$ is not an injective $\mathbb{Z}$-module.
b) If $M$ is divisible and torsionfree, then $M$ is injective. In particular $\mathbb{Q}$ is an injective $\mathbb{Z}$-module.
c) If $R$ is a PID then every divisible module is injective.

Note also that every PID -- in and particular $\mathbb{Z}$ -- is a hereditary ring, so this answers your second question: the submodule $\mathbb{Z}$ of $\mathbb{Q}$ is a noninjective submodule of an injective module over a hereditary ring.

2) (Injective Production Lemma) If $M$ is a flat $R$-module and $N$ is an injective $R$-module, then $\operatorname{Hom}_R(M,N)$ is injective.

This is used to prove that the category of modules over any ring has "enough injectives".

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Examples: Every divisible abelian group ( including those you mentioned ).

No, submodule of injective does not need to be injective. For example Z as a submodule of Q

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$\mathbb{Z}/2\mathbb{Z} \cong (1/2)\mathbb{Z}/\mathbb{Z} \leq \mathbb{Q}/\mathbb{Z}$ gives an example of a submodule of an injective module that is not injective. The rings for which every (left) submodule of an injective (left) module are exactly the (left and right) semisimple artinian rings. These are the rings in which every module is injective. This follows from this existence of injective envelopes. This is often proven using an injective producing lemma (as in this excerpt from Lam's Modules and Rings).

Lam's textbook contains many examples of injective modules on surrounding pages.

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