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Field of fractions

In abstract algebra, the field of fractions (also known as the quotient field) of an integral domain R is the smallest field F that contains R as a subring and in which every nonzero element of R is invertible.^[1]^[2]^[3] It is constructed explicitly as the set of equivalence classes of ordered pairs (a, b) with a, b \in R and b \neq 0, where (a, b) \sim (c, d) if and only if ad = bc.^[1]^[2] The ring operations on these classes are defined by [(a, b)] + [(c, d)] = [(ad + bc, bd)] and [(a, b)] \cdot [(c, d)] = [(ac, bd)], yielding a field with additive identity [(0, 1)] and multiplicative identity [(1, 1)].^[1]^[2]^[3] There exists an injective ring homomorphism \phi: R \to F given by \phi(a) = [(a, 1)], which embeds R into F as a subring.^[1]^[2] This embedding preserves addition and multiplication, ensuring that F extends the algebraic structure of R while introducing inverses for all nonzero elements via [(a, b)]^{-1} = [(b, a)] for a \neq 0.^[3] The field of fractions satisfies a universal property: if K is any field containing an injective ring homomorphism from R, then there exists a unique ring homomorphism \psi: F \to K extending the embedding of R into K.^[1]^[3] Consequently, F is unique up to isomorphism as the "freest" field extension of R in this sense.^[3] Prominent examples include the rational numbers \mathbb{Q}, which form the field of fractions of the integers \mathbb{Z}, and the field of rational functions k(x), the field of fractions of the polynomial ring k over a field k.^[2] These constructions are fundamental in commutative algebra, enabling the study of ideals, localization, and extensions in more general rings, as they provide a way to "invert" elements formally without leaving the domain's structure.^[1]^[2]

Core Concepts

Definition

In abstract algebra, an integral domain is defined as a commutative ring with a multiplicative identity element (unity) that possesses no zero divisors, meaning that if the product of two non-zero elements is zero, then at least one of them must be zero. This structure ensures that multiplication behaves predictably without "accidental" cancellations, distinguishing integral domains from more general rings. Examples include the integers \mathbb{Z} and polynomial rings over fields, but not all integral domains are fields, as they may lack multiplicative inverses for non-units. Given an integral domain R, the field of fractions, denoted \mathrm{Frac}(R) or sometimes K(R), is the smallest field containing R as a subring, obtained by formally inverting all non-zero elements of R.^[3] Equivalently, \mathrm{Frac}(R) is the localization of R at the multiplicative set S = R \setminus \{0\}, which consists of all non-zero elements closed under multiplication and containing the unity.^[4] In this construction, every element of \mathrm{Frac}(R) is represented as a formal quotient a/b with a \in R and b \in R \setminus \{0\}, capturing the idea of division within the domain.^[2] This notion arises historically from the motivation to extend the integers \mathbb{Z} to the rational numbers \mathbb{Q} by permitting division by non-zero integers, thereby creating a field where every non-zero element is invertible; the field of fractions generalizes this process to arbitrary integral domains.^[3]

Construction

Given an integral domain R, the field of fractions, denoted \operatorname{Frac}(R), is constructed explicitly as the set of equivalence classes of ordered pairs (a, b) where a \in R and b \in R \setminus \{0\}. These pairs represent formal quotients a/b, and the equivalence relation is defined by (a, b) \sim (c, d) if and only if ad = bc./18:_Integral_Domains/18.01:_Fields_of_Fractions) This relation is reflexive since aa = aa, symmetric because if ad = bc then cb = da, and transitive as follows: if (a, b) \sim (c, d) and (c, d) \sim (e, f), then ad = bc and cf = de, so multiplying the first by f and the second by b yields adf = bcf = bde. Thus, adf = bde, or d(af - be) = 0. Since d \neq 0 and R has no zero divisors, af = be, so (a, b) \sim (e, f)./18:_Integral_Domains/18.01:_Fields_of_Fractions) The operations on these equivalence classes, denoted [a, b], are defined to mimic fraction arithmetic and are well-defined independent of representatives, again relying on the absence of zero divisors in R. Addition is given by

[a, b] + [c, d] = [ad + bc, bd],

which preserves equivalence: if (a', b') \sim (a, b) and (c', d') \sim (c, d), then a'b = a b' and c'd = c d', so cross-multiplying shows (a'd' + c'b')(b d) = (a d + b c)(b' d') using the relations and no zero divisors to avoid inconsistencies./18:_Integral_Domains/18.01:_Fields_of_Fractions) Multiplication is

[a, b] \cdot [c, d] = [ac, bd],

similarly well-defined since if representatives change equivalently, the products align via a' c' b d = a c b' d'./18:_Integral_Domains/18.01:_Fields_of_Fractions) The additive identity is [0, 1], as [a, b] + [0, 1] = [a \cdot 1 + b \cdot 0, b \cdot 1] = [a, b], and the multiplicative identity is [1, 1], since [a, b] \cdot [1, 1] = [a \cdot 1, b \cdot 1] = [a, b]./18:_Integral_Domains/18.01:_Fields_of_Fractions) For any nonzero [a, b] (where a \neq 0 implies the class is nonzero, as the kernel of the embedding map r \mapsto [r, 1] is trivial in an integral domain), the additive inverse is [-a, b] and the multiplicative inverse is [b, a], verified by [a, b] \cdot [b, a] = [a b, b a] = [1, 1] under the equivalence./18:_Integral_Domains/18.01:_Fields_of_Fractions) This structure (\operatorname{Frac}(R), +, \cdot) forms a field: it is a commutative ring with unity because addition and multiplication inherit associativity, commutativity, and distributivity from R (e.g., ([a, b] + [c, d]) + [e, f] = [ (ad + bc) f + e (b d), b d f ] simplifies to the triple sum via common denominator, matching the direct sum), and every nonzero element has a multiplicative inverse as constructed./18:_Integral_Domains/18.01:_Fields_of_Fractions) The map \iota: R \to \operatorname{Frac}(R) given by \iota(r) = [r, 1] embeds R injectively into the field, confirming it extends R to allow division by nonzero elements./18:_Integral_Domains/18.01:_Fields_of_Fractions)

Properties and Embeddings

Algebraic Properties

The field of fractions of an integral domain R, denoted \operatorname{Frac}(R), is constructed as the set of equivalence classes of pairs (r, s) where r \in R and s \in R \setminus \{0\}, with (r, s) \sim (r', s') if and only if r s' = r' s.^[5] The ring operations are defined by [(r, s)] + [(r', s')] = [(r s' + r' s, s s')] and [(r, s)] \cdot [(r', s')] = [(r r', s s')], where [ \cdot ] denotes the equivalence class.^[6] To verify that \operatorname{Frac}(R) is a field, first note that it forms a commutative ring with unity: addition and multiplication are associative and commutative due to the properties of R, distributivity holds by direct computation, the additive identity is [ (0, 1) ], and the multiplicative identity is [ (1, 1) ].^[5] Moreover, \operatorname{Frac}(R) has no zero divisors, as R is an integral domain: if [(r, s)] \cdot [(r', s')] = [(0, 1)], then r r' = 0 in R, so either r = 0 or r' = 0, implying one of the factors is zero.^[7] Every non-zero element [(r, s)] (with r \neq 0) is invertible, with inverse [(s, r)], since [(r, s)] \cdot [(s, r)] = [(r s, s r)] = [(1, 1)] as r s = s r in R.^[6] Thus, \operatorname{Frac}(R) is a commutative ring with unity in which every non-zero element has a multiplicative inverse, confirming it is a field.^[5] The canonical embedding \iota: R \to \operatorname{Frac}(R) given by \iota(r) = [(r, 1)] is a ring homomorphism, as it preserves addition and multiplication: \iota(r + r') = [(r + r', 1)] = [(r, 1)] + [(r', 1)] and similarly for products.^[7] This embedding is injective because R has no zero divisors: if \iota(r) = [(0, 1)], then (r, 1) \sim (0, 1), so r \cdot 1 = 0 \cdot 1 implies r = 0.^[5] Consequently, R is isomorphic to its image \iota(R), which is a subring of \operatorname{Frac}(R). Every element of \operatorname{Frac}(R) can be represented as r/s with r \in R and s \in R \setminus \{0\}, corresponding to the class [(r, s)]. This representation is unique up to equivalence, as distinct pairs not related by \sim yield different classes, ensuring a well-defined structure without redundancy.^[6] The characteristic of \operatorname{Frac}(R) equals that of R: if \operatorname{char}(R) = n > 0, then n \cdot 1_R = 0 implies n \cdot \iota(1_R) = \iota(n \cdot 1_R) = \iota(0) = 0 in \operatorname{Frac}(R), and n is minimal by injectivity of \iota; if \operatorname{char}(R) = 0, then \operatorname{char}(\operatorname{Frac}(R)) = 0 similarly.^[7] Finally, \operatorname{Frac}(R) is the smallest field containing an isomorphic copy of R as a subring, in the sense that any field extension of R must contain a subfield isomorphic to \operatorname{Frac}(R).^[6]

Universal Property

The field of fractions of an integral domain R, denoted \operatorname{Frac}(R), satisfies the following universal property: for any field F and any injective ring homomorphism \phi: R \to F, there exists a unique field homomorphism \psi: \operatorname{Frac}(R) \to F such that the diagram

\begin{CD} R @>{\phi}>> F\\ @V{i}VV @| \\ \operatorname{Frac}(R) @>{\psi}>> F \end{CD}

commutes, where i: R \to \operatorname{Frac}(R) is the canonical embedding sending r \mapsto (r,1).^[8]^[9] To see this, define \psi on equivalence classes by \psi(a/b) = \phi(a) \cdot \phi(b)^{-1}, where a,b \in R with b \neq 0. This is well-defined because if a/b = a'/b' in \operatorname{Frac}(R), then a b' = b a', so \phi(a) \phi(b') = \phi(b) \phi(a'); since \phi is injective and b \neq 0, \phi(b) \neq 0, ensuring \phi(b)^{-1} exists in F, and the equality \phi(a) \phi(b')^{-1} = \phi(a') \phi(b)^{-1} holds. It is routine to verify that \psi is a ring homomorphism, \psi \circ i = \phi, and that any such homomorphism must take this form, ensuring uniqueness.^[8] This universal property implies that \operatorname{Frac}(R) is unique up to unique isomorphism among all fields containing an isomorphic copy of R: if K is another field with an injective homomorphism \iota: R \to K, then there is a unique isomorphism \operatorname{Frac}(R) \cong K over R if K is generated by the image of R in this way, but more generally, it characterizes \operatorname{Frac}(R) as the initial object in the category of fields equipped with a homomorphism from R.^[9] In categorical terms, the construction of the field of fractions defines a functor \operatorname{Frac}: \mathbf{IntDom} \to \mathbf{Fields} from the category of integral domains (with injective ring homomorphisms as morphisms) to the category of fields (with ring homomorphisms), which is left adjoint to the forgetful functor U: \mathbf{Fields} \to \mathbf{IntDom} that views fields as integral domains; the unit of the adjunction is the canonical embedding i: R \to \operatorname{Frac}(R), and the bijection \operatorname{Hom}_{\mathbf{IntDom}}(R, U(F)) \cong \operatorname{Hom}_{\mathbf{Fields}}(\operatorname{Frac}(R), F) is witnessed precisely by the universal property.^[10]

Examples

Rational Numbers from Integers

The ring of integers \mathbb{Z} is an integral domain, as it is a commutative ring with multiplicative identity $1and possesses no zero divisors: for anya, b \in \mathbb{Z}witha \neq 0andb \neq 0, the product ab \neq 0$./16:_Rings/16.04:_Integral_Domains_and_Fields) The field of fractions of \mathbb{Z}, denoted \operatorname{Frac}(\mathbb{Z}), is the field of rational numbers \mathbb{Q}./18:_Integral_Domains/18.01:_Fields_of_Fractions) Its elements are equivalence classes of ordered pairs (p, q) where p \in \mathbb{Z}, q \in \mathbb{Z} \setminus \{0\}, under the relation (p, q) \sim (r, s) if and only if ps = rq.^[3] These classes are typically represented as fractions \frac{p}{q} in lowest terms, meaning \gcd(p, q) = 1 after adjusting signs so that q > 0, ensuring a unique canonical form for each rational number.^[11] The field operations on \mathbb{Q} are defined as follows: for \frac{p}{q}, \frac{r}{s} \in \mathbb{Q},

\frac{p}{q} + \frac{r}{s} = \frac{ps + rq}{qs},

\frac{p}{q} \cdot \frac{r}{s} = \frac{pr}{qs}.

These operations are well-defined on equivalence classes and satisfy the field axioms, with additive identity \frac{0}{1} and multiplicative identity \frac{1}{1}./18:_Integral_Domains/18.01:_Fields_of_Fractions)^[3] There is a natural embedding i: \mathbb{Z} \to \mathbb{Q} given by i(n) = \frac{n}{1} for n \in \mathbb{Z}, which is an injective ring homomorphism preserving addition and multiplication: i(m + n) = i(m) + i(n) and i(mn) = i(m) \cdot i(n)./18:_Integral_Domains/18.01:_Fields_of_Fractions) This embedding identifies \mathbb{Z} as a subring of \mathbb{Q}, allowing integers to be viewed within the larger field where division by nonzero elements is possible. The construction of \mathbb{Q} from \mathbb{Z} fills the "gaps" left by integer division, enabling denser approximations in the real line; indeed, \mathbb{Q} is dense in the real numbers \mathbb{R}, meaning that between any two reals there exists a rational, which motivates the completion of \mathbb{Q} to the complete ordered field \mathbb{R} via Cauchy sequences or Dedekind cuts.^[12]

Rational Function Field from Polynomials

The polynomial ring k, where k is a field, forms an integral domain, as it is a commutative ring with unity and no zero divisors: the product of two nonzero polynomials is nonzero, since the field k has no zero divisors and the leading coefficients ensure the degree adds under multiplication.^[13] Moreover, k is a Euclidean domain under the degree function as the Euclidean norm, allowing the Euclidean algorithm for gcd computations, which reinforces its lack of zero divisors and enables unique factorization up to units.^[14] The field of fractions of k, denoted \operatorname{Frac}(k) or k(x), consists of equivalence classes of fractions f/g where f, g \in k and g \neq 0, with f/g \sim f'/g' if f g' - f' g = 0.^[15] Elements are considered up to multiplication by units in k^\times, the nonzero elements of k, allowing normalization such that the leading coefficient of the denominator is 1. Addition is defined by f/g + h/i = (f i + h g)/(g i), and multiplication by (f/g) \cdot (h/i) = (f h)/(g i), preserving the field structure since k is an integral domain.^[16] A key tool for manipulating elements in k(x) is partial fraction decomposition, which expresses any f/g with \deg f < \deg g as a sum \sum A_i / (x - r_i)^{m_i} + terms for higher-degree irreducible factors, facilitating integration, simplification, and analysis over k.^[17] The natural embedding k \hookrightarrow k(x) maps each polynomial f to f/1, making k(x) the smallest field containing k as a subring.^[15] For a nonzero rational function f/g in reduced form, the degree is defined as \deg(f/g) = \deg f - \deg g, which determines behavior at infinity and pole orders; for example, if \deg(f/g) < 0, it vanishes at infinity. In algebraic geometry, the rational function field k(x) serves as the function field of the projective line \mathbb{P}^1_k, parametrizing meromorphic functions on this curve, with applications to divisor theory and morphisms between varieties.^[18] Over the complex numbers, \mathbb{C}(x) corresponds to the field of meromorphic functions on the Riemann sphere, where rational functions extend continuously to the compactification \mathbb{P}^1(\mathbb{C}), enabling the study of residues, poles, and uniformization.^[19]

Generalizations

Localization

In commutative algebra, the localization of an integral domain R at a multiplicative subset S \subseteq R generalizes the construction of the field of fractions by inverting only the elements of S, rather than all non-zero elements of R. A multiplicative subset S is a subset containing the multiplicative identity $1 \in R, closed under multiplication, and excluding zero (i.e., $0 \notin S).^[20]^[21] This allows for the formation of a ring S^{-1}R where elements of S become units, while preserving the integral domain structure under appropriate conditions. The ring S^{-1}R is constructed as the set of equivalence classes of pairs (a, s) with a \in R and s \in S, where (a, s) \sim (b, t) if and only if there exists u \in S such that u(at - bs) = 0. For integral domains, where there are no zero divisors, this equivalence simplifies to at = bs, mirroring the standard fraction equivalence.^[20]^[21] The ring operations are defined componentwise: addition by \frac{a}{s} + \frac{b}{t} = \frac{at + bs}{st} and multiplication by \frac{a}{s} \cdot \frac{b}{t} = \frac{ab}{st}, with denominators restricted to elements of S. There is a natural ring homomorphism \phi: R \to S^{-1}R given by \phi(a) = \frac{a}{1}, which is injective if S consists of non-zero-divisors.^[20]^[21] When S = R \setminus \{0\}, the localization S^{-1}R recovers the full field of fractions \operatorname{Frac}(R), as all non-zero elements are inverted.^[20]^[21] A particularly important case is localization at the complement of a prime ideal: if \mathfrak{p} is a prime ideal of R, set S = R \setminus \mathfrak{p}. Then S^{-1}R, denoted R_{\mathfrak{p}}, is a local ring with unique maximal ideal \mathfrak{p} S^{-1}R = \{ \frac{a}{s} \mid a \in \mathfrak{p}, s \in S \}. The prime ideals of R_{\mathfrak{p}} correspond bijectively to the prime ideals of R contained in \mathfrak{p}, facilitating the study of local properties around \mathfrak{p}.^[20]^[21] The localization satisfies a universal property: for any ring homomorphism f: R \to B to a ring B such that f(s) is a unit in B for all s \in S, there exists a unique ring homomorphism \overline{f}: S^{-1}R \to B making the diagram commute, i.e., f = \overline{f} \circ \phi. This property characterizes S^{-1}R as the universal ring extension of R inverting precisely the elements of S.^[20]^[21]

Total Quotient Ring

In commutative algebra, for a commutative ring R with identity (not necessarily an integral domain), the total quotient ring Q(R) is the localization of R at the multiplicative set S consisting of all regular elements of R (i.e., the non-zero-divisors).^[22]^[23] It is constructed as the set of equivalence classes of pairs (a, s) with a \in R and s \in S, where (a, s) \sim (b, t) if and only if there exists u \in S such that u (a t - b s) = 0. The ring operations are defined by \frac{a}{s} + \frac{b}{t} = \frac{a t + b s}{s t} and \frac{a}{s} \cdot \frac{b}{t} = \frac{a b}{s t}.^[22]^[23] The canonical ring homomorphism \phi: R \to Q(R) given by r \mapsto \frac{r}{1} is well-defined, and \phi is injective if and only if R is reduced, meaning R has no nonzero nilpotent elements; in the general case, the image \phi(R) is a subring of Q(R).^[24]^[23] When R is an integral domain, this construction yields Q(R) = \mathrm{Frac}(R), the field of fractions of R.^[22] For reduced rings with finitely many minimal prime ideals, Q(R) is isomorphic to the direct product \prod_{\mathfrak{p}} \operatorname{Frac}(R / \mathfrak{p}), where the product is over the minimal primes \mathfrak{p} of R; in particular, for artinian reduced rings, the minimal primes coincide with the maximal ideals, so Q(R) \cong \prod_{m} R_m.^[22]^[25] A concrete example occurs with R = \mathbb{Z}/6\mathbb{Z}, whose maximal ideals are the principal ideals generated by the images of 2 and 3; the corresponding localizations are R_{(2)} \cong \mathbb{Z}/2\mathbb{Z} and R_{(3)} \cong \mathbb{Z}/3\mathbb{Z}, so Q(R) \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}.^[26]^[27] Unlike the field of fractions, Q(R) is generally not a field, as it is a product of rings whenever R has multiple minimal prime ideals; moreover, the presence of zero divisors in R is reflected in Q(R), where such elements map to tuples with zero components in certain components of the decomposition, thereby detecting the zero divisors through the structure.^[28]

Semifield of Fractions

A commutative semiring is an algebraic structure consisting of a commutative additive monoid (R, +), a multiplicative monoid (R, \cdot) that distributes over addition, and a multiplicative identity $1, but without requiring additive inverses or that every nonzero element is cancellative under multiplication.^[29] For a multiplicatively cancellative commutative semiring R—meaning that for all a, b, c \in R, if a \cdot b = a \cdot c and a \neq 0, then b = c, and similarly for right cancellation—the semifield of fractions \operatorname{Frac}(R) is constructed by localizing at the regular elements, which are the non-zero-divisors of R.^[30] Elements of \operatorname{Frac}(R) are equivalence classes of pairs (a, b) with a \in R and b \in R a regular element, where (a, b) \sim (c, d) if and only if a \cdot d = b \cdot c.^[31] The operations on \operatorname{Frac}(R) are defined componentwise to preserve the semiring structure, without introducing subtraction: for representatives (a, b) and (c, d),

(a, b) + (c, d) = (a \cdot d + b \cdot c, b \cdot d),

(a, b) \cdot (c, d) = (a \cdot c, b \cdot d).

These make \operatorname{Frac}(R) into a semifield, a commutative semiring in which every nonzero element has a multiplicative inverse, given by (a, b)^{-1} = (b, a) for a \neq 0.^[29] There is a natural embedding R \to \operatorname{Frac}(R) sending r \mapsto (r, 1), which is injective due to cancellativity.^[30] When R is an integral domain (a commutative ring with no zero divisors), this construction recovers the classical field of fractions \operatorname{Frac}(R), as the additive inverses present in rings align with the semifield operations.^[31] The multiplicative structure of \operatorname{Frac}(R) forms an abelian group under the inherited multiplication, while the additive structure is a semimodule over \operatorname{Frac}(R) itself, reflecting the absence of negatives.^[29] This setup allows \operatorname{Frac}(R) to model phenomena where subtraction is absent, such as in optimization or positive systems. A representative example is the semiring R = \mathbb{N}_0 = \{0, 1, 2, \dots \} of non-negative integers under standard addition and multiplication, which is multiplicatively cancellative. Here, \operatorname{Frac}(R) consists of equivalence classes (a, b) with a, b \in \mathbb{N}_0, b > 0, yielding the semiring of non-negative rational numbers \mathbb{Q}_{\geq 0} under the usual operations (with $0 included as (0, 1)).^[30] Nonzero elements have multiplicative inverses, but addition lacks subtraction, distinguishing it from the full field of rationals. Another example arises in tropical geometry: consider the max-plus semiring R = (\mathbb{R}_{\geq 0} \cup \{\infty\}, \max, +), where addition is the maximum operation and multiplication is standard addition (with \infty as the additive identity). This is multiplicatively cancellative, and its semifield of fractions \operatorname{Frac}(R) is the tropical semifield (\mathbb{R} \cup \{\infty, -\infty\}, \max, +), embedding R while extending to negative values via inverses (the inverse of x > 0 is -x).^[29] This structure is foundational in tropical algebraic geometry for modeling piecewise-linear phenomena.^[30]

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[PDF] tropical scheme theory - UT Math
Semifield of fractions: Every (multiplicatively) cancellative semiring R embeds into its semifield of frac- tions, denoted Frac(R). ~. : If R is a ring then ...
[28]
[PDF] SEMIRING CONGRUENCES AND TROPICAL GEOMETRY
Mar 20, 2016 · We recall that a cancellative semiring R embeds into its semifield of fractions Frac(R). The elements of Frac(R) are the equivalence classes in ...
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https://web.ma.utexas.edu/users/sampayne/pdf/Math648Lecture5.pdf