## Injective Modules (Pt. III)

**Point of Post: **This is a continuation of this post.

**Divisibility and Injectiveness**

We would now like to discuss how the use of injective modules becomes easier if we insist on using only PIDs (such as ). What is going to end up happen is that while things aren’t quite as simple as projective modules over PIDs (they are just the free ones) PIDs allow us to give a fairly nice description of injective modules over . Namely, if is an integral domain call a left -module *divisible *if for any and there exists such that . To start to understand why it makes sense that injective modules and divisible modules should be related suppose for a second that is a left -module. Then, for any and we have a well-defined -map defined by . This is well-defined precisely because is an integral domain. Now, if is injective we can extend this to a map and we then note that and so divides . Thus, if is injective it is necessarily divisible.

The key theorem is that the converse is true if we assume that is a PID. Why does this make sense? Well thanks to Baer’s criterion we really only have to worry about extending maps from ideals of , but if we assume that is a PID we know that all of these ideals are singly generated. So, we have some map where is a divisible -module. Since is divisible we know we are able to find such that . Define then where . Clearly this is an -map and . Thus, we have proven:

**Theorem: ***Let be a PID. Then, an -module is injective if and only if it’s divisible.*

It’s fairly clear that both quotients and direct sums of divisible -modules are divisible and so we may conclude the following:

**Theorem: ***Let be a PID. **The quotient of injective -modules is injective. Moreover, the direct limit of injective -modules is injective.*

Thus, we now know a lot of examples of injective abelian groups. Namely, if is any characteristic zero field (or any abelian group capable of supporting such a structure) then is an injective abelian group.

**Enough Injectives**

I’d like to finish by proving that categories of modules have “enough injectives” which means that they all modules sit inside an injective one. This will be useful later when we try to approximate arbitrary modules by injective ones leading to the notion of derived functors.

So, we’ve already (secretly) proven that every abelian group sits inside an injective group. Indeed, we merely take any abelian group and find a surjection (this is possible because the category of abelian groups has enough projectives in the form of free modules!). We know then that if is the kernel of this map that . Now, since (being the direct sum of divisible(injective) modules is divisible(injective) and quotients of divisible(injective) modules are divisible(injective)) we are done.

We next note the following fact:

**Theorem: ***Let be a divisible abelian group and a left -module. Then, with is an injective left -module.*

**Proof: **We must show that is an exact functor. But, by adjointness of Hom and tensor we have that but this is clearly just isomorphic to (since is naturally isomorphic to the identity functor) and since is divisible we know that is exact.

We can now prove our desired theorem:

**Theorem(Enough Injectives): ***Let be a ring and a left -module. Then, there exists an injective left -module and a monomorphism .*

**Proof: **We start by embedding into by . This is clearly an -map and is injective by considering .

We know that we can find an embedding of abelian groups where is some divisible abelian group. Then, since the functor is left exact we get an injection . If we can verify that is an -map then will be the desired embedding of into an injective left -module.

To see that this is true let and be arbitrary. We need to show that , but this is simple (I leave it to you).

**References:**

[1] Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. *Advanced Modern Algebra*. Providence, RI: American Mathematical Society, 2010. Print.

[3] Blyth, T. S. *Module Theory.* Clarendon, 1990. Print.

[4] Lang, Serge. *Algebra*. Reading, MA: Addison-Wesley Pub., 1965. Print.

[5] Grillet, Pierre A. *Abstract Algebra*. New York: Springer, 2007. Print.

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