Field of fractions

Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra 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, and the construction is sometimes also called the fraction field, field of quotients, or quotient field ofAll 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.Given an integral domain, we define an equivalence relation onWe denote the equivalence class ofThis notion of equivalence is motivated by the rational numbers, which have the same property with respect to the underlying ringThen the field of fractions is the setwith addition given by and multiplication given by One may check that these operations are well-defined and that, for any integral domain, the multiplicative inverse of(the equivalence class is independent of the choiceThe field of fractions ofis characterized by the following universal property: There is a categorical interpretation of this construction.be the category of integral domains and injective ring maps.to the category of fields that takes every integral domain to its fraction field and every homomorphism to the induced map on fields (which exists by the universal property) is the left adjoint of the inclusion functor from the category of fields toThus the category of fields (which is a full subcategory) is a reflective subcategory ofA multiplicative identity is not required for the role of the integral domain; this construction can be applied to any nonzero commutative rngwith no nonzero zero divisors.is the commutative ring consisting of fractions withTwo special cases of this are notable: Note that it is permitted forThe semifield of fractions of a commutative semiring in which every nonzero element is (multiplicatively) cancellative is the smallest semifield in which it can be embedded.(Note that, unlike the case of rings, a semiring with no zero divisors can still have nonzero elements that are not cancellative.denote the tropical semiring and letThe elements of the semifield of fractions of the commutative semiringare equivalence classes written as with
Quotient ringAlgebraic structureRing theorySubringsFractional idealTotal ring of fractionsProduct of ringsFree product of associative algebrasTensor product of algebrasRing homomorphismsFrobenius endomorphismAlgebraic structuresModuleAssociative algebraGraded ringInvolutive ringCategory of ringsInitial ringTerminal ringFinite fieldNon-associative ringLie ringJordan ringSemiringSemifieldCommutative algebraCommutative ringsIntegral domainIntegrally closed domainGCD domainUnique factorization domainPrincipal ideal domainEuclidean domainPolynomial ringFormal power series ringAlgebraic number theoryAlgebraic number fieldIntegers modulo nRing of integersp-adic integersp-adic numbersNoncommutative algebraNoncommutative ringsDivision ringSemiprimitive ringSimple ringCommutatorNoncommutative algebraic geometryFree algebraClifford algebraGeometric algebraOperator algebraabstract algebraembeddedintegersrational numbersquotient of a ring by an idealcommutative ringlocalizationequivalence relationequivalence classuniversal propertyinjectivering homomorphismcategoricalcategoryfunctorcategory of fieldsleft adjointinclusion functorreflective subcategorymultiplicative identitynonzerozero divisorsrationalsGaussian integersGaussian rationalsisomorphicconvolution space of operatorsDirac delta functiondifferential operatorintegral operatorLaplace transformLocalization (commutative algebra)multiplicative setfractionsprime idealzero-divisorstotal quotient ringtrivial ringcommutative semiringtropical semiringequivalence classesOre conditionOpenStax