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 ${\displaystyle R}$ is sometimes denoted by ${\displaystyle \operatorname {Frac} (R)}$ or ${\displaystyle \operatorname {Quot} (R)}$, and the construction is sometimes also called the fraction field, field of quotients, or quotient field of ${\displaystyle R}$. 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 which is not an integral domain, the analogous construction is called the localization or ring of quotients.

Definition

Given an integral domain ${\displaystyle R}$ and letting ${\displaystyle R^{*}=R\setminus \{0\}}$, we define an equivalence relation on ${\displaystyle R\times R^{*}}$ by letting ${\displaystyle (n,d)\sim (m,b)}$ whenever ${\displaystyle nb=md}$. We denote the equivalence class of ${\displaystyle (n,d)}$ by ${\displaystyle {\frac {n}{d}}}$. This notion of equivalence is motivated by the rational numbers ${\displaystyle \mathbb {Q} }$, which have the same property with respect to the underlying ring ${\displaystyle \mathbb {Z} }$ of integers.

Then the field of fractions is the set ${\displaystyle {\text{Frac}}(R)=(R\times R^{*})/\sim }$ with addition given by

${\displaystyle {\frac {n}{d}}+{\frac {m}{b}}={\frac {nb+md}{db}}}$

and multiplication given by

${\displaystyle {\frac {n}{d}}\cdot {\frac {m}{b}}={\frac {nm}{db}}.}$

One may check that these operations are well-defined and that, for any integral domain ${\displaystyle R}$, ${\displaystyle {\text{Frac}}(R)}$ is indeed a field. In particular, for ${\displaystyle n,d\neq 0}$, the multiplicative inverse of ${\displaystyle {\frac {n}{d}}}$ is as expected: ${\displaystyle {\frac {d}{n}}\cdot {\frac {n}{d}}=1}$.

The embedding of ${\displaystyle R}$ in ${\displaystyle \operatorname {Frac} (R)}$ maps each ${\displaystyle n}$ in ${\displaystyle R}$ to the fraction ${\displaystyle {\frac {en}{e}}}$ for any nonzero ${\displaystyle e\in R}$ (the equivalence class is independent of the choice ${\displaystyle e}$). This is modeled on the identity ${\displaystyle {\frac {n}{1}}=n}$.

The field of fractions of ${\displaystyle R}$ is characterized by the following universal property:

if ${\displaystyle h:R\to F}$ is an injective ring homomorphism from ${\displaystyle R}$ into a field ${\displaystyle F}$, then there exists a unique ring homomorphism ${\displaystyle g:\operatorname {Frac} (R)\to F}$ which extends ${\displaystyle h}$.

There is a categorical interpretation of this construction. Let ${\displaystyle \mathbf {C} }$ be the category of integral domains and injective ring maps. The functor from ${\displaystyle \mathbf {C} }$ to the category of fields which 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 to ${\displaystyle \mathbf {C} }$. Thus the category of fields (which is a full subcategory) is a reflective subcategory of ${\displaystyle \mathbf {C} }$.

A multiplicative identity is not required for the role of the integral domain; this construction can be applied to any nonzero commutative rng ${\displaystyle R}$ with no nonzero zero divisors. The embedding is given by ${\displaystyle r\mapsto {\frac {rs}{s}}}$ for any nonzero ${\displaystyle s\in R}$.^[1]

Examples

• The field of fractions of the ring of integers is the field of rationals: ${\displaystyle \mathbb {Q} =\operatorname {Frac} (\mathbb {Z} )}$.
• Let ${\displaystyle R:=\{a+b\mathrm {i} \mid a,b\in \mathbb {Z} \}}$ be the ring of Gaussian integers. Then ${\displaystyle \operatorname {Frac} (R)=\{c+d\mathrm {i} \mid c,d\in \mathbb {Q} \}}$, the field of Gaussian rationals.
• The field of fractions of a field is canonically isomorphic to the field itself.
• Given a field ${\displaystyle K}$, the field of fractions of the polynomial ring in one indeterminate ${\displaystyle K[X]}$ (which is an integral domain), is called the field of rational functions, field of rational fractions, or field of rational expressions^[2][3][4][5] and is denoted ${\displaystyle K(X)}$.
• The field of fractions of the convolution ring of half-line functions yields a space of operators, including the Dirac delta function, differential operator, and integral operator. This construction gives an alternate representation of the Laplace transform which does not depend explicitly on an integral transform.^[6]

Generalizations

See also

• Ore condition; condition related to constructing fractions in the noncommutative case.
• Projective line over a ring; alternative structure not limited to integral domains.
• Total ring of fractions

References