Given an arbitrary integral domain $R$, the fraction field $Q$ is the smallest field conaining $R$.
In the integers $Q = \mathbb{Q}$
For a polynomial ring $P = {a_nx^n+\dots+a_1}$ the fraction field $Q$ is such that:
$q \in Q \implies q=\frac{p_1}{p_2}: p_1,p_2 \in R$
Is it correct to say that to construct the fraction field from an arbitrary integral domain $R$ we do
$Q = R\times R$ with modified $+,\circ$ operations? and that $R\times R$ is homomorphic to $R$?