What is the motivation behind definition of localization of rings?
From where does the term "localization" come from?
Why is the equivalence relation between the ordered pairs $(m,u),(m',u')$ with $ m,m' \in M$ and $u,u' \in U$ is defined as $(m,u)\sim (m',u')$ if there exists $v\in U$ such that $v(u'm-um')=0$?
Here $M$ is an $R$-module and $U$ is multiplicatively closed subset of the ring $R$.
Best Answer
The term "localization" is indeed taken from geometry. Consider the ring $R = C(\mathbb{R})$ (continuous real valued function on the real numbers) and the set $U$ of function that don't vanish at the origin. Clearly $U$ is multiplicative. Now, what is the geometric interpretation of $U^{-1}R$ ? If we are only interested in the behavior of functions near the origin, a function that doesn't vanish at the origin doesn't vanish in some open neighborhood of the origin. Therefore, by restricting such function to a small enough neighborhood, we get a non-vanishing function and therefore it is possible to take its (multiplicative) inverse. Thus, $U^{-1}R$ can be thought of as the result of concentrating attention to small neighborhoods of the origin and hence the name "localization".
In a general commutative ring we adopt the intuition of the geometric case and think of elements of the ring as "functions" on some "geometric object" whose points are the prime ideals of the ring. The full picture of this is given by the modern algebro-geometric concept of a scheme.