Theorem. Let be a ring and let be a direct system of left -modules over some partially ordered index set Let and for every let be the natural injection map, i.e. where for all Define the -submodule of by
Proof. For every define the -module homomorphism by Clearly if then and thus
So Suppose now that there exists a left -module and -module homomorphisms such that whenever We need to prove that there exists a unique -module homomorphism such that for all Let’s see how this must be defined. Well, we have Now, since every element of is in the form we have to define in this form: The only thing left is to prove that is well-defined. To do so, we define the homomorphism by So I only need to prove that So suppose that and let Then
because for all
Remark 1. The direct limit of a direct system of abelian groups always exists by the theorem because abelian groups are just -modules.
We now show that if is a directed set, then will look much simpler.
Remark 2. If, in the theorem, is directed, then
Proof. Let be any element of Since is directed and is finite, there exists some such that for all Then for every we’ll have
So if we let then
Remark 3. If, in the theorem, is directed and is a direct system of rings, then is also a ring.
Proof. Since are all abelian groups, exists and it is an abelian group by the theorem. So we just need to define a multiplication on Let By Remark 2 there exist such that Choose with Then by the definition of
and thus Similarly Now we define
We need to show that the multiplication we’ve defined is well-defined. So suppose that and is another way of representing Choose such that Then
But we also have
Plugging (2) and (3) into (1) will give us
which proves that the multiplication we defined is well-defined.