Prove that $B[x] \cap B[x^{-1}]$ is integral over $B$
Solution 1:
Since $a\in B[x]\cap B[x^{-1}]$ there exist two non-zero polynomials $f,g\in B[T]$ such that $a=f(x)$ and $a=g(x^{-1})$. Set $m=\deg f$ and $n=\deg g$. Let $M$ be the $B$-submodule of $A$ generated by $1,x,\dots,x^{m+n-1}$. Then $aM\subseteq M$. Furthermore, $M$ is a faithful $B[a]$-module ($1\in M$), and thus we can conclude that $a$ is integral over $B$.