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