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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.
A local ring is Henselian if and only if every finite ring extension is a product of local rings. A Henselian local ring is called strictly Henselian if its residue field is separably closed . By abuse of terminology , a field K {\displaystyle K} with valuation v {\displaystyle v} is said to be Henselian if its valuation ring is Henselian.
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.
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]
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 ...
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 ...