The term localization originates 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 which are non-zero at p and localizes R with respect to S. The resulting ring R* contains only information about the behavior of V near p. Cf. the example given at local ring.
In case R is an integral domain there is an easy construction of the localization. Since the only ring in which 0 is a unit is the trivial ring {0}, the localization R* is {0} if 0 is in S. Otherwise, the field of fractions K of R can be used: we take R* to be the subring of K consisting of the elements of the form the r/s with r in R and s in S. In this case the homomorphism from R to R* is the standard embedding and is injective: but that will not be the case in general. See for example dyadic fraction, for the case R the integers and S the powers of 2.
For general commutative rings, we don't have a field of fractions. Nevertheless, a localization can be constructed consisting of "fractions" with denominators coming from S; in contrast with the integral domain case, one can now safely 'cancel' from numerator and denominator only elements of S.
This construction proceeds as follows: on R × S define an equivalence relation ~ by setting (r_{1},s_{1}) ~ (r_{2},s_{2}) iff there exists t in S such that t(r_{1}s_{2} - r_{2}s_{1}) = 0. We think of the equivalence class of (r,s) as the "fraction" r/s, and using this intuition, the set of equivalence classes R* can be turned into a ring; the map j : R → R* which maps r to the equivalence class of (r,1) is then a ring homomorphism.
The above mentioned universal property is the following: the ring homomorphism j : R → R* maps every element of S to a unit in R*, and if f : R → T is some other ring homomorphism which maps every element of S to a unit in T, then there exists a unique ring homomorphism g : R* → T such that f = g o j.
For example the ring Z/nZ where n is composite is not a integral domain. When n is a prime power it is a finite local ring, and its elements are either units or nilpotent. This implies it can be localized only to a zero ring. But when n can be factorised as ab with a and b coprime and greater than 1, then Z/nZ is by the Chinese remainder theorem isomorphic to Z/aZ × Z/bZ. It we take S to consist only of (1,0) and 1 = (1,1), then the corresponding localization is Z/aZ.
Two classes of localizations occur commonly in commutative algebra and algebraic geometry:
Localizing non-commutative rings is more difficult; the localization does not exist for every set S of prospective units. One condition which ensures that the localization exists is the Ore condition.
One case for non-commutative rings where localization has a clear interest is for rings of differential operators. It has the interpretation, for example, of adjoining a formal inverse D^{-1} for a differentiation operator D. This is done in many contexts in methods for differential equations. There is now a large mathematical theory about it, named microlocalization, connecting with numerous other branches. The micro- tag is to do with connections with Fourier theory, in particular.