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In mathematics, specifically algebraic geometry and its applications, localization is a way of studying an algebraic object "at" a prime. One may study an object by studying it at every prime (the "local question"), then piecing these together to understand the original object (the "local-to-global question").
The above examples of localization of R-modules is abstracted in the following definition. In this shape, it applies in many more examples, some of which are sketched below. Given a category C and some class W of morphisms in C, the localization C[W −1] is another category which is obtained by inverting all the morphisms in W.
The localization of a commutative ring R by a multiplicatively closed set S is a new ring whose elements are fractions with numerators in R and denominators in S.. If the ring is an integral domain the construction generalizes and follows closely that of the field of fractions, and, in particular, that of the rational numbers as the field of fractions of the integers.
The question of when this happens is rather subtle: for example, for the localization of k[x, y, z]/(x 2 + y 3 + z 5) at the prime ideal (x, y, z), both the local ring and its completion are UFDs, but in the apparently similar example of the localization of k[x, y, z]/(x 2 + y 3 + z 7) at the prime ideal (x, y, z) the local ring is a UFD but ...
For commutative rings, ideas of algebraic geometry make it natural to take a "small neighborhood" of a ring to be the localization at a prime ideal. In which case, a property is said to be local if it can be detected from the local rings. For instance, being a flat module over a commutative ring is a local property, but being a free module is not.
The embedding of in maps each in to the fraction for any nonzero (the equivalence class is independent of the choice ). This is modeled on the identity n 1 = n {\displaystyle {\frac {n}{1}}=n} . The field of fractions of R {\displaystyle R} is characterized by the following universal property :
The left Bousfield localization model structure, as described above, is known to exist in various situations, provided that C is a set: . M is left proper (i.e., the pushout of a weak equivalence along a cofibration is again a weak equivalence) and combinatorial
The ring Z/6Z is reduced, however Z/4Z is not reduced: the class 2 + 4Z is nilpotent. In general, Z/nZ is reduced if and only if n = 0 or n is square-free. If R is a commutative ring and N is its nilradical, then the quotient ring R/N is reduced.