Quotient of polynomial ring in two variables is a PID

Question

With $K$ a field and $K[x,y]$ the polynomial ring over it in two variables, the quotient ring of it over the ideal generated by $1-xy$ is a PID.

I've tried using Noetherianess but haven't gotten further than showing it's a UFD.

Answer 1

It is proved in this answer that $K[X,Y]/(XY-1)\simeq K[X,X^{-1}]$ which is a ring of fractions of $K[X]$, so a PID.