What is the canonical ring homomorphism $\varphi: R\to S^{-1}R$?

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What is the canonical ring homomorphism $\varphi: R\to S^{-1}R$ ? Is it $$\varphi(r)=r\ \ ?$$

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It is sending every element of $R$ into the trivial fraction with denominator $1$, i.e $r \mapsto \dfrac r 1$.