Rings with same quotient field Let $R$ be an integral domain and $0 \neq I$ an ideal of $R$. Denote by $\phi: R \rightarrow R/I$ the canonical homomorphism. Let $S$ be a subring of $R/I$ such that $R/I$ is integral over $S$. Suppose $T=\phi^{-1}(S)$. Is it true that the quotient field of $T$ is the same as the quotient field of $R$? I do not know to prove or to disprove the claim. I know it is true if $R \subset Quot(T)$, where $Quot(T)$ is the quotient field of $T$, but I do not think this is true in general.

We do not need the integral assumption:

Take $0 \
eq f \in I$. We have $\phi(rf)=0 \in S$, hence $rf \in T$ for all $r \in R$. In particular $f \in T$. Now we have $r = \frac{rf}{f} \in Quot(T)$, which shows $R \subset Quot(T)$, which is literally all we need.