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

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

thanks!

Printable View

- Nov 7th 2008, 06:05 PMroporteIdeals 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! - Nov 7th 2008, 07:16 PMNonCommAlg
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 $\displaystyle y \in J$:

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