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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 Hilbert polynomial is defined to be the numerical polynomial that agrees with the function n ↦ dim K F n M. for large n. The dimension d(M) of an A n (K)-module M is defined to be the degree of the Hilbert polynomial. It is bounded by the Bernstein inequality. n ≤ d(M) ≤ 2n.
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 ...
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]
Let R be a ring that is graded by the ordered semigroup of non-negative integers, and let + denote the ideal generated by positively graded elements. Then if M is a graded module over R for which M i = 0 {\displaystyle M_{i}=0} for i sufficiently negative (in particular, if M is finitely generated and R does not contain elements of negative ...
Just as the polynomial ring in n variables with coefficients in the commutative ring R is the free commutative R-algebra of rank n, the noncommutative polynomial ring in n variables with coefficients in the commutative ring R is the free associative, unital R-algebra on n generators, which is noncommutative when n > 1.
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 ...
Given a linear operator T on a finite-dimensional vector space V, one can consider the vector space with operator as a module over the polynomial ring in one variable R = K[T], as in the structure theorem for finitely generated modules over a principal ideal domain. Then the spectrum of K[T] (as a ring) equals the spectrum of T (as an operator).