Exercise 5, § 1

Let $A$ be a commutative ring, $B$ a commutative ring containing $A$ and $x$ an invertible element of $B$. Show that every element of $A[x] \cap A[x^{-1}]$ is integral over $A$. (If $y = a_0 + \ldots + a_p x^{-p} = b_0 + \ldots + b_q x^q$, where the $a_i$ and $b_j$ are in $A$, show that the sub-A-module of $B$ generated by $1,\ x,\ \ldots,\ x^{p+q+1}$ is a faithful $A[y]$-module.)