Prove that ideal generated by.... Is a monomial ideal Similar questions have come up on the last few past exam papers and I don't know how to solve it. Any help would be greatly appreciated.. 

Prove that the ideal of $\mathbb{Q}[X,Y]$ generated by $X^2(1+Y^3), Y^3(1-X^2), X^4$ and $ Y^6$ is a monomial ideal.


Let $$\alpha_1=x^2+x^2y^3, \quad\alpha_2=x^4, \quad \beta_1=y^3-x^2y^3, \quad\beta_2=y^6.$$ Then, $(\alpha_1,\alpha_2,\beta_1,\beta_2)\subset (x^2,y^3)$ is easy. The containment $\supset$ is due to the following expressions: 
$$x^2=\alpha_1-x^2\beta_1-y^3\alpha_2$$ $$y^3=\beta_1+y^3\alpha_1-x^2\beta_2.$$
