Hello!
I'd like to show that the ring $\displaystyle K[X,Y] / (XY -1)$ is isomorphic to $\displaystyle S^{-1}k[X] $ where S is the multiplicative system of powers of X.
Thanks for any comment!
By modding out $\displaystyle K[X,Y]$ by $\displaystyle XY-1$ we are saying that in the new ring, $\displaystyle XY-1=0$ or rather $\displaystyle XY=1$. Therefore Y is the inverse of X.
In $\displaystyle S^{-1}k[X]$, we are adding in $\displaystyle X^{-1}$ to $\displaystyle k[X]$ and all it's powers.
From here the isomorphism should be fairly clear