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Math Help - ismomorphism between ... (conmutative algebra)

  1. #1
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    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!
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  2. #2
    Member Haven's Avatar
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    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
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  3. #3
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    Re: ismomorphism between ... (conmutative algebra)

    Quote Originally Posted by Haven View Post
    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
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  4. #4
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    Re: ismomorphism between ... (conmutative algebra)

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