Fraction field

Let $R$ be an integral domain . The fraction field $\mathrm{Frac}(R)$ is the set of equivalence classes of pairs $(a,b)\in R\times R$ with $b\neq 0$, under

Write the class of $(a,b)$ as $a/b$. Addition and multiplication are defined by

and these operations make $\mathrm{Frac}(R)$ a field . The map $R\hookrightarrow \mathrm{Frac}(R)$, $a\mapsto a/1$, is an injective ring map.

Universal property. If $K$ is a field and $\iota:R\to K$ is a ring monomorphism , then there exists a unique field homomorphism $\tilde\iota:\mathrm{Frac}(R)\to K$ with $\tilde\iota(a/1)=\iota(a)$ for all $a\in R$.

For domains, $\mathrm{Frac}(R)$ agrees with the total ring of fractions (since every nonzero element is a non-zero-divisor).