algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled...

(UK); and the fraction bar, solidus, or fraction slash. In typography, fractions stacked vertically are also known as en or nut fractions, and diagonal...

Field-flow fractionation, abbreviated FFF, is a separation technique invented by J. Calvin Giddings. The technique is based on separation of colloidal...

total quotient ring or total ring of fractions is a construction that generalizes the notion of the field of fractions of an integral domain to commutative...

fields Fp. Given an integral domain R, its field of fractions Q(R) is built with the fractions of two elements of R exactly as Q is constructed from the integers...

embed it in its field of fractions.) The archetypical example is the ring Z {\displaystyle \mathbb {Z} } of all integers. Every field is an integral domain...

to polynomials over the field of fractions of a unique factorization domain. This makes essentially equivalent the problems of computing greatest common...

algebraic geometry they are elements of some quotient ring's field of fractions. In complex geometry the objects of study are complex analytic varieties...

over its field of fractions F, typically the field of the rational numbers, and we denote R[X] and F[X] the rings of polynomials in a set of variables...

given subset S of R. If S is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes...

Partial fraction decomposition Partial fractions in integration Function field of an algebraic variety Algebraic fractions – a generalization of rational...

In mathematics, an overring of an integral domain contains the integral domain, and the integral domain's field of fractions contains the overring. Overrings...

is a field extension of the field of fractions of A. If A is a subring of a field K, then the integral closure of A in K is the intersection of all valuation...

every non-zero element x of its field of fractions F, at least one of x or x−1 belongs to D. Given a field F, if D is a subring of F such that either x or...

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...

field of rational fractions in s {\displaystyle s} over K {\displaystyle K} . The notation L / K is purely formal and does not imply the formation of...

numbers. In the case of coefficients in a unique factorization domain R, "rational numbers" must be replaced by "field of fractions of R". This implies that...

Two fractions are added as follows: a b + c d = a d + b c b d . {\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.} If both fractions are...

uses a fraction slash "⁠ / {\displaystyle /} ⁠".) Quotient rings are distinct from the so-called "quotient field", or field of fractions, of an integral...

The notion of irreducible fraction generalizes to the field of fractions of any unique factorization domain: any element of such a field can be written...

properly containing it. There is some discrete valuation ν on the field of fractions K of R such that R = {0} ∪ {\displaystyle \cup } {x ∈ {\displaystyle...

[t]\!]} (it is the field of fractions of the formal power series ring k [ [ t ] ] . {\displaystyle k[\![t]\!].} ) The function field of an algebraic variety...

field of fractions of R (the field of rational numbers, if R is the integers). This second definition is not used in this article. The equivalence of...

to be the field of fractions of the affine coordinate ring of any open affine subset, since all such subsets are dense. There are a number of formal similarities...

{\displaystyle K[X]} , i.e. the field of rational functions on K {\displaystyle K} , by the universal property of the field of fractions. We can conclude that in...

continued fractions, we can distinguish three cases: The two sequences {Τ2n−1} and {Τ2n} might themselves define two convergent continued fractions that have...

field of fractions, and let P be a non-zero prime ideal of R. Then, the localization of R at P, denoted RP, is a principal ideal domain whose field of...

construction of a field of fractions, or more generally localization of a ring. The right Ore condition for a multiplicative subset S of a ring R is that...

consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. The importance of the...

the Krull dimension of the ring A; and if A is an integral domain, d is also the transcendence degree of the field of fractions of A over k. The theorem...