Let $X \subset Y$ be a subring of ring $Y$, and suppose that $Y$ is integral over $X$. Show that $Y^* \cap X=X^*$. Can the integrality condition be omitted?
Let $x \in X^*$, since $X \subset Y$, $x \in Y^*$. Conversely, let $y \in Y^* \cap X$ and $y^{-1} \in Y$. Since $Y$ is integral over $X$, there exists $r_1,..,r_n \in X$ such that $$y^{-n}+ r_1y^{-n+1}+...+r_n=0$$ As $y \in X$, $y^{-1}=-(r_1+r_2y+..+r_ny^{n-1}) \in X$. Hence $Y^* \cap X=X^*$.
Could anyone indicate the counterexamples that integrality condition can not be omitted? Are there any examples in number ring?