## Direct Limit of Rings (Pt. III)

**Point of Post: **This is a continuation of this post.

*Examples of Direct Limits (cont.)*

We now come to our final example. What we claim is that the localization is the direct limit of the ‘s over this directed system. The put-in maps are just the inclusions (which are trivially well-defined). What we now need to prove is that whenever . To see this we merely note that if then

where the step is true since, now being in the full , we may “cancel” the whereas we couldn’t before (you can’t have denominators of the form in !). What we must now show is that is universal with respect to this property. Namely, suppose that is some set of ring homomorphisms (where is some given ring) such that for all . We then define by the rule . To prove that this map is well-defined we merely note that if then there exists some such that thus

Clearly this is a ring homomorphism and Moreover, it’s clear that this was the only such map that would have worked since if is another such map then we see that for any one has that

*General Construction***Theorem:**

*Let be a preordered set and be a directed system of rings over . Then, if is the ideal*

*of (where are the canonical injections) then is a direct limit of this directed system.*

**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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