Integrally closed domain.

Question

Suppose $A$ is a unique factorization domain, $a$ is an element of $A$. Is the ring $A[x,a/x]$ always integrally closed? ($x$ is a variable over $A$)

Thanks!

Answer 1

Let $A$ be a normal domain, $a\in A$, then $B=A[x,a/x]$ is a normal domain. In fact, let $y=a/x$, then $B=A[x,1/x]\cap A[y,1/y]$. The ring $A[x,1/x]$ is a localization of the normal domain $A[x]$, thus normal. So, $B$ is normal.