Field of fractions

2020 Mathematics Subject Classification: Primary: 13-XX [MSN][ZBL]

of an integral domain $R$; quotient field, field of quotients

The smallest field containing the integral domain $R$, considered as fractions with elements of $R$ as numerator and denominator. The construction generalises the construction of the rational numbers from the ring of integers.

Let $S$ denote $R \setminus {0}$: since $R$ is an integral domain, $S$ is closed under multiplication. Define an equivalence $\sim$ on $R \times S$ by $$ (x,y) \sim (u,v) \Leftrightarrow xv = yv \ . $$ We denote the equivalence class of $(x,y)$ by $x/y$ and the quotient by $F$. Operations are defined on $F$ by $$ x/y + u/v = (xv+yu)/yv $$ and $$ x/y \cdot u/v = (xu)/(yv) \ . $$ These definitions are compatible with the relation $\sim$, that is, do not depend on the choice of representative of the equivalence classes.

It may be verified that $F$ becomes a field under these operations, and the map from $R$ to $F$ by $x \mapsto x/1$ is an embedding. The field of fractions has the universal property that if $R$ embeds in a field $K$ then the embedding extends to an embedding of $F$ into $K$.

In the opposite direction, given a field $F$, every subring $R$ of $F$ is necessarily an integral domain. We can identify the field of fractions of $R$ with the set $\{ ab^{-1} : a \in R, b \in R\setminus\{0\}\}$ as a subfield of $F$.

See Fractions, ring of for more general constructions.

References

• P.M. Cohn, Skew Field Constructions, London Mathematical Society lecture note series 27, Cambridge University Press (1977) ISBN 0-521-21497-1