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Prime ideals in $\mathbb{C}[x,y]$ were listed here; they are:

(i) $(0)$.

(ii) $(f)$, where $f$ is an irreducible polynomial.

(iii) $(x-\lambda,y-\mu)$, where $\lambda,\mu \in \mathbb{C}$.

Now let $R \subseteq \mathbb{C}[x,y]$, with $\dim(R)=2$.

Question. Is it possible to find a somewhat similar list for the prime ideals in $R$?

If this question is too general, then let us concentrate on the following three special cases:

Let $a,b,c,d \in \mathbb{C}[x,y]$, with each two of $\{a,b,c,d\}$ algebraically independent over $\mathbb{C}$; for example: $a=x^2,b=y^2,c=xy,d=x-y$.

(1) $R=\mathbb{C}[a,b]$. In this case, if I am not wrong, $R$ is isomorphic to $\mathbb{C}[x,y]$, so the same result holds with $x,y$ replaced by $a,b$.

(2) $R=\mathbb{C}[a,b,c]$.

(3) $R=\mathbb{C}[a,b,c,d]$.

Any hints and comments are welcome! Thank you.

I have asked the above question in MSE.

Edit: Special cases answers are also welcome.

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  • $\begingroup$ What is the algebraic properties of $R$? $\endgroup$
    – Eduardo Longa
    Mar 16, 2021 at 3:37
  • $\begingroup$ $R$ could be any subring of $\mathbb{C}[x,y]$ having dimension two. For example, I am not assuming that $R$ is integrally closed in its field of fractions (=normal) or that the field of fractions of $R$ is $\mathbb{C}(x,y)$. However, if it is possible to answer my question in some special cases, it would be nice to know. Thank you. $\endgroup$
    – user237522
    Mar 16, 2021 at 4:01
  • $\begingroup$ Also, I am not assuming that $R \subseteq \mathbb{C}[x,y]$ is an integral extension or flat. $\endgroup$
    – user237522
    Mar 16, 2021 at 4:07

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