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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.
More generally, if F is a local ring and n is a positive integer, then the quotient ring F[X]/(X n) is local with maximal ideal consisting of the classes of polynomials with constant term belonging to the maximal ideal of F, since one can use a geometric series to invert all other polynomials modulo X n.
Such a ring is necessarily a reduced ring, [5] and this is sometimes included in the definition. In general, if A is a Noetherian ring whose localizations at maximal ideals are all domains, then A is a finite product of domains. [6] In particular if A is a Noetherian, normal ring, then the domains in the product are integrally closed domains. [7]
In particular if k is a field, the ring of integers, or a principal ideal domain, then the polynomial ring [, …,] is regular. In the case of a field, this is Hilbert's syzygy theorem. Any localization of a regular ring is regular as well. A regular ring is reduced [b] but need not be an integral domain. For example, the product of two regular ...
On the other hand, the formal power series ring over a UFD need not be a UFD, even if the UFD is local. For example, if R is the localization of k[x, y, z]/(x 2 + y 3 + z 7) at the prime ideal (x, y, z) then R is a local ring that is a UFD, but the formal power series ring R[[X]] over R is not a UFD.
This ring can also be described as the coordinate ring of the cuspidal cubic curve y 2 = x 3 over K. The subring K[t 3, t 4, t 5] of the polynomial ring K[t], or its localization or completion at t=0, is a 1-dimensional domain which is Cohen–Macaulay but not Gorenstein. Rational singularities over a field of characteristic zero are Cohen ...
The ring of Laurent polynomials [,] is an extension of the polynomial ring [] obtained by "inverting ". More rigorously, it is the localization of the polynomial ring in the multiplicative set consisting of the non-negative powers of . Many properties of the Laurent polynomial ring follow from the general properties of localization.
As the latter formulation makes clear, a polynomial ring is the group algebra over a vector space, and writing in terms of corresponds to choosing a basis for the vector space. Then an ideal I, or equivalently a module R / I , {\displaystyle R/I,} is a cyclic representation of R (cyclic meaning generated by 1 element as an R -module; this ...