## Direct Limit of Rings (Pt. I)

**Point of Post: **In this post we discuss the direct limit of rings, prove their universal characterization, and give some examples.

*Motivation*

Unsurprisingly, we can take the direct limit of rings in much the same way we can take the direct limit of modules. This reflects that the notion of a direct limit can truly be defined in any category, though we shall not take that view (yet). All the intuition is going to be the same for that of modules, and so as to not just repeat it verbatim, we omit it in the below material. We shall see that many, many interesting constructions can be described as certain direct limits.

*Directed Systems*

Let be a preordered set, then a *directed system of rings *over is an ordered pair where is a ring for each and the are ring homomorphisms satisfying whenever and .

Let’s now take a look at some examples of directed systems of rings:

Let be some ring and define to be the natural inclusion of into the first coordinates of . Clearly then forms a directed system over the directed set .

Consider a set of rings and consider the trivial preorder on . We then define the morphism to be . Evidently then forms a trivial directed system over the preordered set .

Fix some . Let be the neighborhood system of (i.e. all open sets containing ). Define an ordering on by the opposite inclusion ordering, i.e. if and only if . Define then for in the maps (where denotes the ring of holomorphic functions on the set) to be the restriction map . Evidently then (this really is easy to check, the restriction of a restriction is a restriction of the smaller) forms a directed system over the directed set .

Let be a commutative unital ring and let be a unitally multiplicative subset of . Define the relation on by if and only if (in the usual notion of divisibility). For each let denote the localization of at . There are obvious maps when . Namely, since we know that for some , fixing such an we define . Note that this is in fact independent of which divisor we take. Indeed, suppose that then indeed it suffices by definition to show that but and from where the equality follows. We claim that when . Indeed, suppose that and then and

since evidently for each (since , we may take our representative complementary divisor to be ) we thus have that is a directed system over the (easily verified) directed set .

*Direct Limits*

Now that we have some examples of directed systems of rings, we can define the directed limits over such systems. Namely, given a preordered set and a directed system over this preordered set, we define a *direct limit* of this system to be an ordered pair where is some ring and is a set, called a *limit cone,* of ring homomorphisms satisfying and universal with respect to this property: given any set of ring homomorphisms (where is some ring) such that there exists a unique ring homomorphism satisfying .

Shocking as this may be (not shocking at all, anything defined in terms of universal characterizations ‘morally’ should be) unique up to (unique) isomorphism. Indeed:

**Theorem: ***Let a preordered set, and a directed system over . Then, if and are two direct limits of this direct system, then .*

**Proof: **By virtue of the existence of the maps and satisfying and we afforded by the universal properties of the two direct products maps and such that and . From this we deduce that and . But, since the identity maps and satisfy these two identities respectively we have by uniqueness that and respectively. Thus, is an isomorphism as desired.

**References:**

[1] Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. *Advanced Modern Algebra*. Providence, RI: American Mathematical Society, 2010. Print.

[3] Blyth, T. S. *Module Theory.* Clarendon, 1990. Print.

[4] Lang, Serge. *Algebra*. Reading, MA: Addison-Wesley Pub., 1965. Print.

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