I am following the notes of Gathmann to learn myself about commutative algebra. I have the following exercise written in them (without solutions at the end):
Exercise 1.13. Show that the equation of ideals $$(x^3-x^2,x^2y-x^2,xy-y,y^2-y) = (x^2,y)\cap(x-1,y-1)$$ holds in the polynomial ring $\mathbb{C}[x,y]$. Is this a radical ideal? What is its zero locus in $\mathbb{A}^2_{\mathbb{C}}$?
While the two last questions are easy to solve: it is not radical because $x(x-1)$ is not in the ideal but $(x(x-1))^2 = x^2(x-1)^2 = (x-1)(x^3 - x^2)$, so is in the ideal, and its zero locus is obviously $\{(0,0),(1,1)\}$.
The inclusion $\subseteq$ is easy too, because the product of ideals is included in the intersection. But of the $\supseteq$ part, I suppose that I have an element of the intersection, so a poynomial $p$ such that there exists $p_1,p_2,p_3,p_4 \in \mathbb{C}[x,y]$ such that: $$p = p_1x^2 + p_2y$$ $$p = p_3(x-1) + p_4(y-1)$$
I tried to add them or but them equal, without success. The only thing I can prove is that $p^2$ is in the ideal, by multiplying the two writings, but the ideal is not radical, so it doesn't work either!
Someone can give me a hint or help me please?