Can we find an ideal $I$ s.t. $ k[x,y]/I$ = $k[x^2,y^2,xy]?$

Question

Can we find an ideal $I$ of $k[x,y]$ s.t. $ k[x,y]/I$ = $k[x^2,y^2,xy]?$

Answer 1

Hint. $\dim k[x^2,xy,y^2]=2$ and $\dim k[x,y]/I<2$ (unless $I=0$).