Determine all maximal ideals in the following ring

Question

Determine all maximal ideals in $$R=\frac{\mathbb{C}[x,y]}{(x^3-x^2y,xy^2+xy+x+1)}$$

Attempt: By the correspondence theorem, it suffices to determine all maximal ideals in $\mathbb{C}[x,y]$ containing $(x^3-x^2y,xy^2+xy+x+1)$. In this ideal we have $$x^3-x^2y=0\implies x(x^2-xy)=0$$

If $x=0$, the ideal reduces to $(1)=R$ which is not proper and therefore not maximal.

If $x^2=xy$, then $x=0$ or $x=y$. In the second case, the ideal reduces to $(x^3+x^2+x+1)$. It therefore suffices to determine all linear factors of $x^3+x^2+x+1$ since $\mathbb{C}[x]$ is a PID. Since $x^3+x^2+x+1=(x+1)(x-i)(x+i)$ the maximal ideals in $\mathbb{C}[x,y]$ containing $(x^3-x^2y,xy^2+xy+x+1)$ are given by $$(x+1),(x-i),(x+i)$$

Is this approach correct? Now I realize I took out a $y$, and I probably need to put it back in the ideal somewhere. I am confused on this process of "eliminating the $y$" and then having to put it back in the ideal in the end.

For example, I used the relation $x^3-x^2y=0$ in order to eliminate one of the variables, in order to simplify the process of finding maximal ideals. However, how do I "keep track" of the eliminated variable correctly?

Edit:Should the answer be

$$(x+1,x^3-x^2y),(x+i,x^3-x^2y),(x-i,x^3-x^2y)$$

instead? Do you sort of, have to put back in whatever generator you "zeroed out" or eliminated in the ideal, in the end? I am confused on this process.

Answer 1

Your approach is good. You want to describe maximal ideals in $\mathbb{C}[x,y]$ that contains the given ideal. Any maximal ideal in $\mathbb{C}[x,y]$ is of the form $(x-a,y-b)$ so it suffices to describe possible values for $a$ and $b$. Equivalently, from a geometric point of view, pairs $(a,b)$ of complex numbers satisfying the two equations in the given ideal. Now as you point out, $a^2(a-b)=0$ implies that $a=b$ or $a=0$ and the only possible case is $a=b$. We are left to determine what values of $a$ satisfies $a^3+a^2+a+1=0$. What you conclude is that the maximal ideals containing the given ideal are exactly

$$(y+1,x+1),(y-i,x-i),(y+i,x+i).$$ The image of these under the quotient is the maximal ideals you are looking for.

Note: These are the same ideals as you have written down, just substitute your value for $x$ in the second equation.