§ Injective module
- An injective module is a generalization of the properties of as an abelian group ( module.)
- In particular, given any injective group homomorphism and a morphism , then we induce a group homomorphism , where are abelian groups.
- We can think of this injection as identifying a submodule (subgroup) of .
- Suppose we wish to define the value of at some . If is in the subgroup then define .
- For anything outside the subgroup , we define the value of to be .
- Non-example of injective module: See that this does not work if we replace with .
- Consider the injective map given by Consider the quotient map . We cannot factor the map through as [ for contradiction ]. since any map is determined by where sends the identity. But in this case, the value of . Thus, is not an injective abelian group, since we were unable to factor the homomorphism along the injective .
- Where does non-example break on Q? Let's have the same situation, where we have an injection given by . We also have the quotient map . We want to factor where . This is given by