Let $D$ be a domain. Let $D^{*}$ be the set of non-zero elements of $D$. We define a relation $\sim$ on $D\times D^{*}$ as follows: $(a, b)\sim(c, d)$ if and only if $ad=bs$. It is easily checked that $\sim$ is an equivalence relation. We denote the equivalence classes with respect to this relation as $a/b$. Let $F$ denote the quotient set.
The map $\eta$ from $D$ into $F$ such that every element $a$ in $D$ goes to $a/1$ is a monomorphism.
I wonder to know whether this function is surjective? So for every fraction $a/b$ in $F$ can we find an element $a$ in $D$ such that $\eta({a})=a/b$?
Would you help me please? Thank you in advance.