Adjunction of a root to a UFD Let $R$ be a unique factorization domain which is a finitely generated $\Bbbk$-algebra for an algebraically closed field $\Bbbk$. For $x\in R\setminus\{0\}$, let $y$ be an $n$-th root of $x$. My question is, is the ring
$$ A := R[y] := R[T]/(T^n - x) $$
a unique factorization domain as well?
Edit: I know the classic counterexample $\mathbb{Z}[\sqrt{5}]$, but $\mathbb{Z}$ it does not contain an algebraically closed field. I am wondering if that changes anything.
Edit: As Gerry's Answer shows, this is not true in general. What if $x$ is prime? What if it is a unit?

I think you can get a counterexample to the unit question, even in characteristic zero, and even in an integral domain (in contrast to Georges' example), although there are a few things that need checking. 
Let $R={\bf C}[x,1/(x^2-1)]$, so $1-x^2$ is a unit in $R$. Then $$(1+\sqrt{1-x^2})(1-\sqrt{1-x^2})=xx$$   
It remains to check that 


*

*$R$ is a UFD, 

*$1\pm\sqrt{1-x^2}$ and $x$ are irreducibles in $R[\sqrt{1-x^2}]$ 

*$1\pm\sqrt{1-x^2}$ and $x$ are not associates in $R[\sqrt{1-x^2}]$  
