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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 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.
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").
It is the integers localized at 2. More generally, given any commutative ring R and any prime ideal P of R, the localization of R at P is local; the maximal ideal is the ideal generated by P in this localization; that is, the maximal ideal consists of all elements a/s with a ∈ P and s ∈ R - P.
1 → 2 results immediately from the preservation of integral closure under localization; 2 → 3 is trivial; 3 → 1 results from the preservation of integral closure under localization, the exactness of localization, and the property that an A-module M is zero if and only if its localization with respect to every maximal ideal is zero.
Multiplicity one criterion states: [2] if the completion of a Noetherian local ring A is unimixed (in the sense that there is no embedded prime divisor of the zero ideal and for each minimal prime p, ^ / = ^) and if the multiplicity of A is one, then A is regular. (The converse is always true: the multiplicity of a regular local ring is ...
That is, an element u of a ring R is a unit if there exists v in R such that = =, where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u. [1] [2] The set of units of R forms a group R × under multiplication, called the group of units or unit group of R.
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.