So what exactly is the localization of a ring?
It is creating a field-imitation (and NOT necessarily a field with multiplicative inverses) for every ring. It includes the creation of multiplicative inverse- imitations (for elements that are not zero-divisors).
How does it do this? It apes the steps taken to create a field of fractions from an integral domain. A good description can be found on pg. 60 of Watkins’ “Topics in Commutative Ring Theory”, First Edition.
I would like to concentrate on one small point. Why is it that where belong to the same multiplicative system? This description does not exactly ape the condition for integral domains: . Simply put, why the extra ??
Because we have no means of isolating a regular element that is a common factor!! In every case, including that of integral domains, we get something like , where . Then we say is not a zero divisor. Hence, . Here, we can’t say anything about . Hence, the condition here is that .
Now where does the “multiplicative system” part come in?? That part seems fairly arbitrary, doesn’t it? How does it help prove transitivity? Because you’re going to get something of the form anyway! Hence, if you didn’t add the , how would you justify the ? Simple?
But why “multiplicative system”? Why not a simple for some ? Mainly because this allows us to localize with respect to different multiplicative systems within the ring. It helps us to generalize. We can choose the denominators and (as in ) from subsets of the ring, or the whole ring itself. It is the properties of these subsets that we will explore in the coming paragraphs.
Say we localize with respect to multiplicative system of ring . Then we choose all denominators from . We need to choose denominators from the same multiplicative system because the way fraction multiplication is defined: we want multiplicative closure.
Why choose in from the same multiplicative system as the denominators too? Why choose from a multiplicative system at all? We have to choose from the same multiplicative system because soon we start having products of , and we don’t want them to be from outside of the set we’re choosing the from. This multiplicative system is the same as that of the denominators because denominators are also multiplied to the . Hence, we want algebraic closure.