# Thread: ismomorphism between ... (conmutative algebra)

1. ## ismomorphism between ... (conmutative algebra)

Hello!
I'd like to show that the ring $K[X,Y] / (XY -1)$ is isomorphic to $S^{-1}k[X]$ where S is the multiplicative system of powers of X.

Thanks for any comment!

2. ## Re: ismomorphism between ... (conmutative algebra)

By modding out $K[X,Y]$ by $XY-1$ we are saying that in the new ring, $XY-1=0$ or rather $XY=1$. Therefore Y is the inverse of X.

In $S^{-1}k[X]$, we are adding in $X^{-1}$ to $k[X]$ and all it's powers.

From here the isomorphism should be fairly clear

3. ## Re: ismomorphism between ... (conmutative algebra)

Originally Posted by Haven
By modding out $K[X,Y]$ by $XY-1$ we are saying that in the new ring, $XY-1=0$ or rather $XY=1$. Therefore Y is the inverse of X.

In $S^{-1}k[X]$, we are adding in $X^{-1}$ to $k[X]$ and all it's powers.

From here the isomorphism should be fairly clear
I understand everything you say but I can't see the ismorphism... :S

4. ## Re: ismomorphism between ... (conmutative algebra)

Try the homomorphism:
$X \mapsto X$ and $Y \mapsto X^{-1}$
See if you can show it's bijective.