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Thread: Ideals over K[x,y]

  1. #1
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    Ideals over K[x,y]

    Prove the equality between ideals in K[x,y]:

    $\displaystyle <x+xy, y+xy,x^2,y^2>=<x,y>$

    thanks!
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  2. #2
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    Quote Originally Posted by roporte View Post
    Prove the equality between ideals in K[x,y]:

    $\displaystyle <x+xy, y+xy,x^2,y^2>=<x,y>$

    thanks!
    call the ideals in the LHS and RHS $\displaystyle I$ and $\displaystyle J$ respectively. clearly $\displaystyle I \subseteq J.$ so we only need to show that $\displaystyle x \in I$ and $\displaystyle y \in J$:

    $\displaystyle x=x+xy + x^2y - x(y+xy) \in J, \ \ y=y+xy + y^2x - y(x+xy) \in J. \ \ \ \Box$
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