"Quotient field" redirects here. Not to be confused with Quotient ring.
Algebraic structure → Ring theory Ring theory
Basic concepts
Rings
• Subrings
• Ideal
• Quotient ring
• Fractional ideal
• Total ring of fractions
• Product of rings
• Free product of associative algebras
• Tensor product of algebras
Ring homomorphisms
• Kernel
• Inner automorphism
• Frobenius endomorphism
Algebraic structures
• Module
• Associative algebra
• Graded ring
• Involutive ring
• Category of rings
• Initial ring
• Terminal ring
Related structures
• Field
• Finite field
• Non-associative ring
• Lie ring
• Jordan ring
• Semiring
• Semifield
Commutative algebra
Commutative rings
• Integral domain
• Integrally closed domain
• GCD domain
• Unique factorization domain
• Principal ideal domain
• Euclidean domain
• Field
• Finite field
• Composition ring
• Polynomial ring
• Formal power series ring
Algebraic number theory
• Algebraic number field
• Ring of integers
• Algebraic independence
• Transcendental number theory
• Transcendence degree
p-adic number theory and decimals
• Direct limit/Inverse limit
• Zero ring
• Integers modulo pn
• Prüfer p-ring
• Base-p circle ring
• Base-p integers
• p-adic rationals
• Base-p real numbers
• p-adic integers
• p-adic numbers
• p-adic solenoid
Algebraic geometry
• Affine variety
Noncommutative algebra
Noncommutative rings
• Division ring
• Semiprimitive ring
• Simple ring
• Commutator
Noncommutative algebraic geometry
Free algebra
Clifford algebra
• Geometric algebra
Operator algebra
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In abstract 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 on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral domain elements.
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. For a commutative ring that is not an integral domain, the analogous construction is called the localization or ring of quotients.
and 24 Related for: Field of fractions information
algebra, the fieldoffractionsof an integral domain is the smallest field in which it can be embedded. The construction of the fieldoffractions 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...
embed it in its fieldoffractions.) The archetypical example is the ring Z {\displaystyle \mathbb {Z} } of all integers. Every field is an integral domain...
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...
codomain is L. The set of rational functions over a field K is a field, the fieldoffractionsof the ring of the polynomial functions over K. A function f...
to polynomials over the fieldoffractionsof a unique factorization domain. This makes essentially equivalent the problems of computing greatest common...
algebraic geometry they are elements of some quotient ring's fieldoffractions. In complex geometry the objects of study are complex analytic varieties...
Field-flow fractionation, abbreviated FFF, is a separation technique invented by J. Calvin Giddings. The technique is based on separation of colloidal...
The notion of irreducible fraction generalizes to the fieldoffractionsof any unique factorization domain: any element of such a field can be written...
is a field extension of the fieldoffractionsof A. If A is a subring of a field K, then the integral closure of A in K is the intersection of all valuation...
Asymmetrical flow field-flow fractionation (AF4) is most versatile and most widely used sub-technique within the family offield flow fractionation (FFF) methods...
every non-zero element x of its fieldoffractions 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...
properly containing it. There is some discrete valuation ν on the fieldoffractions K of R such that R = {0} ∪ {\displaystyle \cup } {x ∈ {\displaystyle...
closure in its fieldoffractions is A itself. Spelled out, this means that if x is an element of the fieldoffractionsof A that is a root of a monic polynomial...
for fieldsof non-zero characteristic. If a simple extension K(s) / K is not finite, the field K(s) is isomorphic to the fieldof rational fractions in...
from the so-called "quotient field", or fieldoffractions, of an integral domain as well as from the more general "rings of quotients" obtained by localization...
total quotient ring or total ring offractions is a construction that generalizes the notion of the fieldoffractionsof an integral domain to commutative...
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...
to be the fieldoffractionsof the affine coordinate ring of any open affine subset, since all such subsets are dense. There are a number of formal similarities...
called the Picard group of R. If R is an integral domain with the fieldoffractions F of R, then there is an exact sequence of groups: 1 → R ∗ → F ∗ →...
{\displaystyle K[X]} , i.e. the fieldof rational functions on K {\displaystyle K} , by the universal property of the fieldoffractions. We can conclude that in...
Global Information
![{\displaystyle \mathbb {Z} [1/p]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab7b4e5c999e46c7614c3264b1d80708f4363433)
