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I'm trying to solve the next problem:

Let $R$ be an integral domain, and $\lbrace S_i\rbrace_i$ a family of multiplicative closed subsets of $R$ such that $R=\bigcap_i S_i^{-1}R$. Then $R$ is integrally closed if and only if each $S_i^{-1}R$ is.

From left to right I'm trying to use the fact that if the field of fractions of $R$ is $K$, the field of fractions of each $S_i^{-1}R$ is isomorphic to $K$, so i want to take an element $r/s\in K$ such that $$(\frac{r}{s})^n+(\frac{r}{s})^{n-1}\frac{a_1}{b_1}+\cdots+\frac{a_n}{b_n}=\frac{0}{1}$$ where $\frac{a_j}{b_j}\in S_i^{-1}R$, and conclude that necesarlly $\frac{r}{s}\in S_i^{-1}R$, but i dont know how to proceed. Thanks if some one can help me.

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    $\begingroup$ Hint. If $x\in K$ is integral over $S^{-1}R$ then there is $s\in S$ such that $sx$ is integral over $R$. $\endgroup$ – user26857 Apr 1 '18 at 6:01
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    $\begingroup$ In your notation those $b_j\in S_i$ for all $j$. Clear the denominators. $\endgroup$ – user26857 Apr 1 '18 at 6:02
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    $\begingroup$ Thanks your hint was perfect. $\endgroup$ – Nicolas Cuervo Apr 2 '18 at 15:50

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