Field of fractions

Definition

Let $R$ be an integral domain. On the set of pairs $(a, b)$ with $a, b \in R$ and $b \neq 0$, let $(a, b) \sim (c, d)$ if $a d = b c$. This is an equivalence relation. Write $a/b$ for the class of $(a, b)$. The field of fractions $\operatorname{Frac}(R)$ is the set of these classes, with the operations $$\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}, \qquad \frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d}.$$ It is a field, and the map $R \to \operatorname{Frac}(R)$, $a \mapsto a/1$, is an injective ring homomorphism.