Show that $k[x,y]/(xy-1)$ 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.

In the lines of the accepted answer, one can show that the global dimension of $k[x,x^{-1}]$ is $1$, so that if it is isomorphic to a polynomial ring, it must be isomorphic to one of the form $D[y]$ with $D$ a semisimple commutative domain, which is thus a field.