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 that is not an integral domain, the analogous construction is called the localization or ring of quotients.

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

and multiplication given by

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: