Ads
related to: field of fractions model
Search results
Results From The WOW.Com Content Network
The field of fractions of an integral domain is sometimes denoted by or (), and the construction is sometimes also called the fraction field, field of quotients, or quotient field of . All four are in common usage, but are not to be confused with the quotient of a ring by an ideal , which is a quite different concept.
The field of fractions F of A is a superreal field if F strictly contains the real numbers , so that F is not order isomorphic to . If the prime ideal P is a maximal ideal, then F is a field of hyperreal numbers (Robinson's hyperreals being a very special case). [citation needed]
Informally, a field is a set, along with two operations defined on that set: an addition operation written as a + b, and a multiplication operation written as a ⋅ b, both of which behave similarly as they behave for rational numbers and real numbers, including the existence of an additive inverse −a for all elements a, and of a multiplicative inverse b −1 for every nonzero element b.
In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in its field of fractions is A itself. Spelled out, this means that if x is an element of the field of fractions of A that is a root of a monic polynomial with coefficients in A, then x is itself an element of A.
There the function field of an algebraic variety V is formed as the field of fractions of the coordinate ring of V (more accurately said, of a Zariski-dense affine open set in V). Its elements f are considered as regular functions in the sense of algebraic geometry on non-empty open sets U , and also may be seen as morphisms to the projective ...
In algebraic geometry, the Néron model (or Néron minimal model, or minimal model) for an abelian variety A K defined over the field of fractions K of a Dedekind domain R is the "push-forward" of A K from Spec(K) to Spec(R), in other words the "best possible" group scheme A R defined over R corresponding to A K.
In abstract algebra, the total quotient ring [1] or total ring of fractions [2] is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings R that may have zero divisors. The construction embeds R in a larger ring, giving every non-zero-divisor of R an inverse in the larger ring.
For example, if = then its function field is isomorphic to () where is an indeterminant and the field is the field of fractions of polynomials in . Then, a place v p {\displaystyle v_{p}} at a point p ∈ X {\displaystyle p\in X} measures the order of vanishing or the order of a pole of a fraction of polynomials p ( x ) / q ( x ) {\displaystyle ...