Here is Problem 7.5.2 in Dummit & Foote:
Let $R$ be an integral domain and let $D$ be a nonempty subset of $R$ that is closed under multiplication. Prove that the ring of fractions $D^{-1} R$ is isomorphic to a subring of the quotient field of $R$(hence is also an integral domain).
And here is Theorem 15:
I believe the only difference is that $D$ in the theorem does not contain zero and has no zero divisors but the one in the problem only is closed under multiplication, so the problem 7.5.2 will be proved by the same way the theorem is proved. Am I correct? (I am still confused about the proof of the uniqueness part in the theorem though)
Consider the ring $D^{-1}R$ then we have $D\subset R\setminus \left\{0\right\}$ and so we have the canonical morphism $$\varphi:R \to \mathrm{Quot}\left(R\right)$$ which by construction is injective and $\varphi\left(d\right)=d$ is invertible as it is not zero and an element in a field. In particular it factors by (2) "uniqueness" via $D^{-1}R\hookrightarrow \mathrm{Quot}\left(R\right)$ and so the image of that monomorphism is isomorphic to $D^{-1}R$ and hence a subring of $\mathrm{Quot}\left(R\right)$.
Short comment: the above is a standard argument using Universal Properties for embeddings/uniqueness up to unique isomorphism. Uniqueness up to unique isomorphism for example arises as you can run the above argument for any multiplicative subset, in particular the full multiplicative halfgroup as well to get $\varphi$ being a unique iso. This is why I do not like their formulation of this as "Uniqueness" and would prefer if they called it "Universal Property"