Ideals over K[x,y]

**roporte** · November 7th 2008, 06:05 PM

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

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

thanks!

**NonCommAlg** · November 7th 2008, 07:16 PM

Originally Posted by roporte

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

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

thanks!

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

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

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