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

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

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!

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

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

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

Originally Posted by Haven
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
I understand everything you say but I can't see the ismorphism... :S

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

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