Let be a ring. In a previous exercise, we constructed the direct limit of abelian groups indexed by a directed, partially ordered set . Verify that if the are left -modules, then is naturally a left -module, that the natural injections are -module homomorphisms, and that satisfies an appropriate universal property.
Let be a nonempty, directed, partially ordered set and let be a family of left -modules indexed by . Suppose that for all with , there exists an -module homomorpism such that (1) whenever and (2) (the identity mapping) for all .
Let , and define a relation on by if and only if there exists such that .
Previously, we showed that is an equivalence relation and that via the operator (where is any element such that ) the set is an abelian group which we denote . Moreover, for each , the mapping given by is a group homomorphism.
Define an action of on by . We claim that this action is well defined and makes into a left -module.
- (Well-defined) Suppose . Then there exists such that . Now , and since the are module homomorphisms, . Thus , and we have . Hence this action is well-defined.
- Now let and let . Let . Then we have .
- Let and . Then .
- Let and let . Note that . Then .
So is a left -module. Further, suppose that has a 1 and the are all unital. Then , so that is unital. Finally, note also that for all , , and , we have . Thus the injections are all -module homomorphisms. Certainly we have for all with .
We claim that , together with the , is universal with respect to this property in the following sense: If is a left -module, and for each , a module homomorphism, such that whenever , then there exists a unique -module homomorphism such that for all . That is, there exists a unique -module homomorphism such that the following diagram commutes for all .
We will first prove existence and then uniqueness.
A group homomorphism given by exists such that for all by the universal property of direct limits of abelian groups. It remains to be seen that is a module homomorphism- that is, that it preserves scalars. To that end, let and let . Then ; so is indeed an -module homomorphism.
For uniqueness, suppose that is another -module homomorphism such that for all . In particular, is a group homomorphism. By the universal property of direct limits of abelian groups, , so that is unique as desired.