$\mathrm{Frac}(R)/R$ as a direct limit Let $R$ be an integral domain, let $F=\mathrm{Frac}(R)$ be its field of fractions. Then $R$ is a submodule of $F$, hence we have quotient module $F/R$.
It is true that
$$F/R \cong \varinjlim R/rR$$ 
where the direct set is the set of nonzero element $r\in R$, and the partial order is given by
$$ r_{1} \preceq r_{2} \Leftrightarrow r_{1}\mid r_{2}$$
Can anyone explain to me that why it is true or not? Thank you very much.

What is actually true is that $F$ is the limit of the direct system $(R_r, u_{rs})$, where each $R_r$ is copy of the ring $R$ and $u_{rs}\colon R_r\longrightarrow R_s$ is defined as multiplication by $r$ in $R$ whenever $r\mid s$.
