Let be a ring. Let be a nonempty partially ordered set, and let be a family of left -modules. In a previous exercise we constructed the inverse limit of the abelian groups . Verify that is naturally a left -module, that the projections are module homomorphisms, and that satisfies an appropriate universal property.
Let be a partially ordered set and let be a family of left -modules. Suppose we have, for all , an -module homomorphism such that (1) whenever and (2) (the identity map) for all .
The inverse limit of the (with respect to the ) is the subset . We showed previously that is an abelian group. In fact, we claim that is a submodule of . To that end, let and ; since whenever , . Moreover, since for all , , so that is nonempty. Thus by the submodule criterion is a left -module as we would expect.
Moreover, for all , we have . Thus the projections are -module homomorphisms. If the are all unital, then is unital since . Certainly we have for all .
We claim that is universal with respect to this property in the following sense. If is a left -module and a family of -module homomorphisms such that, whenever , we have , then there exists a unique -module homomorphism such that for all . That is, there is a unique homomorphism such that the following diagram commutes for all .
By the universal property of inverse limits of abelian groups, there exists a group homomorphism such that for all . Specifically, . It remains to be seen that is a module homomorphism. To that end, let and . Then . So is in fact a module homomorphism.
For uniqueness, suppose is another -module homomorphism such that for all . Then in particular is a group homomorphism, and by the universal property of inverse limits of abelian groups, .