Theorem. Let be a commutative ring, an -module and a direct system of -modules. Then
The above isomorphism is an isomorphism of -modules. We will give the proof of this theorem at the end of this post. So let’s get prepared for the proof!
Notation 2. We proved in here that exists. Recall that, as a part of the definition of direct limit, we also have canonical homomorphisms satisfying the relations for all For modules we explicity defined by (see the theorem in here), but we will not be needing that.
Notation 3. Let be an -module. For all with we will let and By Example 5, is a direct system of -modules and so exists. We also have canonical homomorphisms satisfying for all
Notation 4. For every we will let
Remark 1. Clearly and for all because
Lemma. Let be an -module. Suppose that for every there exists an -module homomorphism such that for all Let Then
1) There exist -module homomorphisms such that for all
2) There exists a unique -module homomorphism such that for all
Proof. 1) are defined very naturally: for all Then
2) This part is obvious from the first part and the universal property of direct limit. (See Definition 1 in here)
Proof of the Theorem. We will show that satisfies the conditions in the definition of (see Definition 1 in here) and thus, by the uniqueness of direct limit, and the theorem is proved. The first condition is satisfied by Remark 1. For the second condition (the universal property), suppose that is an -module and are -module homomorphisms such that whenever So the hypothesis in the above lemma is satisfied. For and let and be the maps in the lemma. Define the map by for all and See that is -bilinear and so it induces an -module homomorphism defined by We also have
for all and Thus So the only thing left is the uniqueness of which is obvious because, as we mentioned in the second part of the above lemma, is uniquely defined for a given
Remark 2. If is not commutative, are left -modules and is a right -module (resp. -bimodule), then the isomorphism in the theorem is an isomorphism of abelian groups (resp. left -modules).