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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.
The skew-polynomial ring is defined similarly for a ring R and a ring endomorphism f of R, by extending the multiplication from the relation X ...
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 ...
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]
Then the localization of R with respect to S is a -graded ring. If I is an ideal in a commutative ring R , then ⨁ n = 0 ∞ I n / I n + 1 {\textstyle \bigoplus _{n=0}^{\infty }I^{n}/I^{n+1}} is a graded ring called the associated graded ring of R along I ; geometrically, it is the coordinate ring of the normal cone along the subvariety ...
A polynomial ring in infinitely many variables: ... is called the localization of R with respect to S. For example, if R is a commutative ring and f an ...
The fact that polynomial rings over a field are Noetherian is called Hilbert's basis theorem. Moreover, many ring constructions preserve the Noetherian property. In particular, if a commutative ring R is Noetherian, the same is true for every polynomial ring over it, and for every quotient ring, localization, or completion of the ring.