In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the quotient field or the field of fractions of the integral domain. The elements of the quotient field of the integral domain R have the form a/b with a and b in R and b ≠ 0. The quotient field of the ring R is sometimes denoted by Quot(R). The quotient field of the ring of integers is the field of rationals, Q = Quot(Z). The quotient field of a field is isomorphic to the field itself. History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...

One can construct the quotient field Quot(R) of the integral domain R as follows: Quot(R) is the set of equivalence classes of pairs (n, d), where n and d are elements of R and d is not 0, and the equivalence relation is: (n, d) is equivalent to (m, b) iff nb=md (we think of the class of (n, d) as the fraction n/d). The embedding is given by n(n, 1). The sum of the equivalence classes of (n, d) and (m, b) is the class of (nb + md, db) and their product is the class of (mn, db). In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Although P iff Q is most standard, common alternative phrases include P is necessary and sufficient for Q and P...

The quotient field of R is characterized by the following universal property: if f : R → F is a ring monomorphism from R into a field F, then there exists a unique ring monomorphism g : Quot(R) → F which extends f. In category theory, abstract algebra and other fields of mathematics, frequently constructions are defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ...

Assigning to every integral domain its quotient field defines a functor from the category of integral domains (with ring monomorphisms as morphisms) to the category of fields. This functor is left adjoint to the forgetful functor which assigns to every field its underlying integral domain. In category theory, a functor is a special type of mapping between categories. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ... A forgetful functor is a type of functor in mathematics. ...