Are these ideals the same? I have already proved that $(X^3-Y^3,X^2Y-X)\subseteq(X^2-Y,X-Y^2)$ since the elements $X^3-Y^3$ and $X^2Y-X $ can be written as a linear combination of $(X^2-Y,X-Y^2)$. 
However, I can't write $X^2-Y,X-Y^2$ as a combination of $(X^3-Y^3,X^2Y-X)$. How can I prove that $(X^2-Y,X-Y^2) \subseteq(X^3-Y^3,X^2Y-X)$ or they are not the same ideals?
Thank you very much.

If $(X^2-Y,X-Y^2) \subseteq(X^3-Y^3,X^2Y-X)$, then $X^2-Y\in (X^3-Y^3,X^2Y-X)$. There exist two polynomials $f,g$ such that $X^2-Y=(X^3-Y^3)f(X,Y)+(X^2Y-X)g(X,Y)$. Send $X$ to $0$ in the equation and find $Y=Y^3f(0,Y)$ $\Leftrightarrow$ $1=Y^2f(0,Y)$. Now send $Y$ to $0$ and obtain $1=0$, a contradiction. 
Conclusion: $(X^3-Y^3,X^2Y-X)\subsetneq(X^2-Y,X-Y^2)$.
