Find all the ideals $\mathfrak{b}$ that fit $\mathfrak{a}\subsetneq\mathfrak{b}\subsetneq\mathbb{R}[X,Y,Z]$,being$\mathfrak{a}=(X^2-2X-4,Y-X^2,Z-X^3)$

Question

Let $\mathbb{K}$ be a field, and $\mathfrak{a}=(X^2-2X-4,Y-X^2,Z-X^3)$ an ideal of $\mathbb{K}[X,Y,Z]$.

1. The first exercise was to see if $\mathfrak{a}$ is prime or maximal in $\mathbb{K}[X,Y,Z]$, when $\mathbb{K}=\mathbb{Q}$ and $\mathbb{K}=\mathbb{R}$. I have done that, and I reach to the conclusion that $\mathfrak{a}$ is both prime and maximal in $\mathbb{Q}[X,Y,Z]$ but it is not maximal neither prime in $\mathbb{R}[X,Y,Z]$.

2. Now, in the second exercise, in the cases where $\mathfrak{a}$ is not maximal, (if I have done it correctly, in $\mathbb{R}[X,Y,Z]$), I have to find all the ideals $\mathfrak{b}$ which fit $\mathfrak{a}\subsetneq\mathfrak{b}\subsetneq\mathbb{K}[X,Y,Z]=\mathbb{R}[X,Y,Z]$.

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I don't know if I'm doing it ok but I've tried to do the following:

If I have understood well:

I need ideals of this type: $(f(X,Y,Z), X^2-2X-4,Y-X^2,Z-X^3)$, which are maximal in $\mathbb{R}[X,Y,Z]$ or what it is the same, I have to find ideals $(f(X,Y,Z),X^2-2X-4)$ which are maximal in $\mathbb{R}$. But now how can I continue?

Am I doing it in a correct way?

Answer 1

In $\mathbb{R}[X,Y,Z]$, you have that $$\begin{align}\mathfrak{a}&=((X-r_1)(X-r_2),Y-2X-4,Z-8X-8)\\&=(X-r_1,Y-2X-4,Z-8X-8)\cap(X-r_2,Y-2X-4,Z-8X-8)\\&=(X-r_1,Y-2r_1-4,Z-8r_1-8)\cap(X-r_2,Y-2r_2-4,Z-8r_2-8)\end{align}$$

with $r_1,r_2=1\pm\sqrt{5}$.

The ideals $\mathfrak{b}_i=(X-r_1,Y-2r_1-4,Z-8r_1-8)$, for $i=1,2$, are maximal and contain $\mathfrak{a}$.