Why do the Localization of a Ring

This question may be a bit vague, but neverthless, i would like to see an answer. Wikipedia tells me that:

• In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring.

It is clear that for integral domains, we have the Field of Fractions, and we work on it. It's obvious that a Field has multiplicative inverses. Now if we consider any arbitrary ring $R$, by the definition of localization, it means that we are adding multiplicative inverses to $R$ thereby wanting $R$ to be a division ring or a field. My question, why is it so important to look at this concept of Localization. Or how would the theory look like if we never had the concept of Localization.

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You don't necessarily get a field or a division ring, because you are not necessarily adding inverses to every element, just to some. –  Arturo Magidin Mar 17 '11 at 21:01
The title is missing a verb? –  Mariano Suárez-Alvarez Mar 17 '11 at 21:11
@Mariano: Or a couple of articles: "Why do the localization of a ring?" –  Arturo Magidin Mar 17 '11 at 21:19

Localization is a technique which allows one to concentrate attention to what is happening near a prime, for example. When you localize at a prime, you have simplified abruptly the behavior of your ring outside that prime but you have more or less kept everything inside it intact.

For lots of questions, this significantly simplifies things.

Indeed, there are very general procedures, in lots of contexts, which go by the name of localization, and their purpose is usually the same: if you are lucky, the problems you are interested in can be solved locally and then the "local solutions" can be glued together to obtain a solution to your original problem. Moreover, an immense deal of effort has been done in order to extent the meaning of "local" so as to be able to apply this strategy in more contexts: I have always loved the way the proofs of some huge theorems of algebraic geometry consist more or less in setting up an elaborate technology in order to be able to say the magical "It is enough to prove this locally", and then, thanks to the fact that we worked so much in that technology, immediately conclude the proof with a "where it is obvious" :)

Of course, all sort of bad things can happen. For example, sometimes the "local solutions" cannot be glued together into a "global solution", &c. (Incidentally, when this happens, so that you can do something locally but not glue the result, you end up with a cohomology theory which, more or less, is the art of dealing with that problem)

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In Algebraic Geometry, if you have a variety $\mathbf{V}$, then you have the corresponding coordianate ring $A(\mathbf{V})$ to study the variety. To each point $p\in\mathbf{V}$ there corresponds a maximal ideal $\mathfrak{M}$ of $A(\mathbf{V})$.

The ring $A(\mathbf{V})$ carries information about what is going on in all of $\mathbf{V}$. If you are only interested in understanding what is happening at (or near) the point $p$, then a lot of the information in $A(\mathbf{V})$ doesn't really matter. If you localize $A(\mathbf{V})$ at $\mathfrak{M}$, what you get is precisely the information about what is going on near $p$. For example, there is a one-to-one correspondence between the prime ideals of this localization and the subvarieties of $\mathbf{V}$ that go through the point $p$.

The coordinate ring is often not an integral domain (e.g., if the variety is not irreducible). So you want to be able to localize in rings that are not integral domains.

More generally, localizing gives you a way to focus on particular parts of the ideal structure of $R$, even when $R$ is not an integral domain (so that there is no ring of fractions to embed into); if the ideal you are looking at is a prime ideal, then you can "strip away" all the stuff that is not inside of $\mathfrak{p}$ and make it "not matter", so that you can just focus on the stuff what is going on inside of $\mathfrak{p}$. This is important and useful even when we don't have a division ring or a field to embed into.

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