Show that $k[x,y]/(xy-1)$ is not isomorphic to a polynomial ring in one variable. 
Let $R=k[x,y]$ be a polynomial ring ($k$, of course, is a field). Show that
  $R/(xy-1)$ is not isomorphic to a polynomial ring in one variable.


Suppose $R\simeq K[T]$, where $K$ is a commutative ring. Then $K$ is an integral domain and since $\dim K[T]=1$ we get $\dim K=0$ (why?). It follows that $K$ is a field. So $k[X,X^{-1}]\simeq K[T]$. Now use this answer.
