# Identifying elements in quotient field of integral domain with elements in the ring (Notation)

Let $R$ be an integral domain.

A result in ring theory says that $R$ can be embedded in its quotient field.

I wish to identity $\frac{rs}{s}$ in the quotient field with $r\in R$.

What is the usual notation to write this? Is $$\frac{rs}{s}=r$$ ok?

I have an idea but the notation is cumbersome: Define $\phi_s: R\to S^{-1}R$ by $$\phi_s(r)=rs/s$$, then $\phi_s$ is a monomorphism. We can then write $\phi_s^{-1}(\frac{rs}{s})=r$.

Thanks for any help.

Well, the definition of equivalence of elements in the quotient field is $\frac{a}{b}\equiv\frac{c}{d}$ if $ad=bc$.
Via this definition, $\frac{r}{1}\equiv\frac{rs}{s}$, and of course we are assuming the identification of $r\mapsto \frac{r}{1}$, a clear isomorphism.