Localization

Equivalence. Define $(a,b)\sim(c,d)\iff ad=bc$. This is an equivalence relation (reflexive, symmetric, transitive, using that $R$ is an integral domain for cancellation).

Field structure. Addition: $\frac ab+\frac cd=\frac{ad+bc}{bd}$. Multiplication: $\frac ab\cdot\frac cd=\frac{ac}{bd}$. These are well-defined. The zero is $\frac01$, the unit is $\frac11$. Every nonzero $\frac ab$ has inverse $\frac ba$.

Embedding. $R\hookrightarrow\operatorname{Frac}(R)$ by $a\mapsto\frac a1$.

Example. $\operatorname{Frac}(\mathbb{Z})=\mathbb{Q}$: every $\frac pq$ with $p,q\in\mathbb{Z}$, $q\neq0$, is a rational number, and every rational is of this form.

Conclusion. Every integral domain embeds in a field.