In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module R, so that it consists of fractions such that the denominator s belongs to a given subset S of R. If S is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field of rational numbers from the ring of integers.
The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf theory. In fact, the term localization originated in algebraic geometry: if R is a ring of functions defined on some geometric object (algebraic variety) V, and one wants to study this variety "locally" near a point p, then one considers the set S of all functions that are not zero at p and localizes R with respect to S. The resulting ring contains information about the behavior of V near p, and excludes information that is not "local", such as the zeros of functions that are outside V (c.f. the example given at local ring).
