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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.

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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.

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