# ismomorphism between ... (conmutative algebra)

• Nov 17th 2013, 11:36 AM
Lolyta
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!
• Nov 24th 2013, 02:37 AM
Haven
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
• Nov 24th 2013, 07:35 AM
Lolyta
Re: ismomorphism between ... (conmutative algebra)
Quote:

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
• Nov 27th 2013, 05:58 PM
Haven
Re: ismomorphism between ... (conmutative algebra)
Try the homomorphism:
$X \mapsto X$ and $Y \mapsto X^{-1}$
See if you can show it's bijective.