Localization (Pt. II)
Point of Post: This post is a continuation of this one.
So, as stated, one of the main reasons localizations is important is the following:
Theorem: Let be a commutative unital ring and a unital multiplicatively closed subset. Then,
Proof: Let , then we see that and so .
In particular, if we do take to be an integral domain and then is a field, which we call the field of fractions of and denote it .
Now, what we’d like to say is that is the ‘smallest’ field containing . In fact, we make the more precise following universal characterization:
Theorem(Universal Characterization of Localizations): Let be a commutative unital ring and a multiplicatively closed subset. Then, given any commutative unital ring and a unital homomorphism such that then factors through . Said differently, there exists a unique morphism such that .
Proof: Clearly such a map, if it exists, is unique. So, to produce such a map we define given by . Let’s first show that this is actually a well-defined map. Note that if then for some . We see then that and so noting that since we have that are units we may conclude from this that and so
To prove that is a morphism we note that evidently , and that
The conclusion follows by noticing that
This gives us that the localization of a ring at a multiplicative subset is ‘unique’ for the properties it possesses, namely:
Theorem: Let and be commutative unital rings and multiplicative. Then, if and only if there exists a unital homomorphism such that , if and only if for some , and every element of is of the form for some and .
Proof: By the first property and the previous universal characterization of we have that there exists a unital homomorphism , the second property says it’s injective since implies which implies that for some and so , and clearly the last characteristic says precisely that is a surjection. It thus follows that is an isomorphism.
As a corollary to this we get the following:
Theorem: Let be an integral domain and a field containing . Then, the subbfield (intersection of all fields) containing is isomorphic to .
Ideal Theory of Localizations
So, we want to figure out, given a commutative unital ring and a unital multiplicative subset what the ideals of the localization look like. Well, as always we have natural maps from to given by and given by . While this doesn’t give us the whole story it’s a start. That said, while it is, at least theoretically, apparent what looks like, it’s not immediately as obvious what looks like. Luckily, and not altogether unexpectedly, there is a nice presentation for this set. Namely
Theorem: Let be a commutative unital ring and a unital multiplicative subset. Then, is equal to
Proof: Evidently this set is an ideal containing and so it suffices to prove that any ideal containing contains this set. To see this we merely note that if is an ideal in containing then it contains for every and , from where the conclusion follows.
 Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.
 Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. Basic Abstract Algebra. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.