Field of fractions

• 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 or field of rational fractions[1][2][3] and is denoted ${\displaystyle K(X)}$.

Let ${\displaystyle R}$ be any integral domain.

For ${\displaystyle n,d\in R}$ with ${\displaystyle d\neq 0}$,

the fraction

${\displaystyle {\frac {n}{d}}}$

denotes the equivalence class of pairs

${\displaystyle (n,d)}$,

where ${\displaystyle (n,d)}$ is equivalent to ${\displaystyle (m,b)}$ if and only if ${\displaystyle nb=md}$.

(The definition of equivalence is modelled on the property of rational numbers that ${\displaystyle {\frac {n}{d}}={\frac {m}{b}}}$ if and only if ${\displaystyle nb=md}$.)

The field of fractions ${\displaystyle \operatorname {Frac} (R)}$ is defined as the set of all such fractions ${\displaystyle {\frac {n}{d}}}$.

The sum of ${\displaystyle {\frac {n}{d}}}$ and ${\displaystyle {\frac {m}{b}}}$ is defined as

${\displaystyle {\frac {nb+md}{db}}}$,

and the product of ${\displaystyle {\frac {n}{d}}}$ and ${\displaystyle {\frac {m}{b}}}$ is defined as

${\displaystyle {\frac {nm}{db}}}$

(one checks that these are well defined).

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 modelled on the identity ${\displaystyle {\frac {n}{1}}=n}$.

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

if ${\displaystyle h:R\rightarrow F}$ is an injective ring homomorphism from ${\displaystyle R}$ into a field ${\displaystyle F}$,

then there exists a unique ring homomorphism ${\displaystyle g:\operatorname {Frac} (R)\rightarrow F}$ which extends ${\displaystyle h}$.

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

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 ${\displaystyle s\in R}$.[4]

• Ore condition; this is the condition one needs to consider in the noncommutative case.
• Projective line over a ring; alternative structure not limited to integral domains.