example of injective module
In the category of unitary -modules (which is the category of Abelian groups), every divisible Group is injective, i.e. every Group such that for any and , there is a such that . For example, and are divisible, and therefore injective.
We have to show that, if is a divisible Group, is any homomorphism, and is a subgroup of a Group , there is a homomorphism such that the restriction . In other words, we want to extend to a homomorphism .
Let be the set of pairs such that is a subgroup of containing and is a homomorphism with . Then ist non-empty since it contains , and it is partially ordered by
For any ascending chain
in , the pair is in , and it is an upper bound for this chain. Therefore, by Zorn’s Lemma, contains a maximal element .
It remains to show that . Suppose the opposite, and let . Let denote the subgroup of generated by . If , the sum is in fact a direct sum, and we can extend to by choosing an arbitrary image of in and extending linearly. This contradicts the maximality of .
Let us therefore suppose contains an element , with minimal. Since , and is defined on , exists, and furthermore, since is divisible, there is a such that . It is now easy to see that we can extend to by defining , in contradiction to the maximality of .
Therefore, . This proves the statement. ∎
|Title||example of injective module|
|Date of creation||2013-03-22 17:43:40|
|Last modified on||2013-03-22 17:43:40|
|Last modified by||Glotzfrosch (19314)|