Let be a ring and be a prime ideal.
is a multiplicative subset of .
In this case, the localization is called (somewhat confusingly) the localization of at the prime and given the notation .
Geometric interpretation
Let for some .
Let be a maximal ideal.
The correspondence between algebra (rings, modules, and algebras of functions) and geometry (affine varieties, etc.) can be summarized by the following map:
where is the maximal ideal of all functions which are zero at the point .
Therefore, looking at elements in means we are looking at functions where which means it is a polynomial that is not zero at the point .
In this way, we have made inverses for the functions that are “invertible” at the point since we can be sure to not divide by 0.
Note that we can still evaluate functions in at the point since we can take .
Thus, we get a neighborhood around , but is not defined “far” away from since we are not guaranteed to have for .
So gives information about local behaviors of things around the point that relates to .