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This question is based on an exercise in Artin's Algebra:

Which ideals in the polynomial ring $R:={Bbb C}[x,y]$ contain $f_1=x^2+y^2-5$ and $f_2=xy-2$?

Using Hilbert's (weak) nullstellensatz, one can identify all the maximal ideals of $R$ that contain $f_1$ and $f_2$. For the general ideals contain $(f_1, f_2)$, it suffices to identify ideals in $R/(f_1,f_2)$ by the correspondence theorem. But I don't see how to go on. Is there a systematic way to do it?

I don't know if this counts as a "systematic" way of doing it or not, but you can do the following.

First, in the ring $R$ you have $y = 2/x$, so you have the isomorphism $$R = frac{mathbb{C}[x,y]}{(f_1,f_2)}cong frac{mathbb{C}[x,x^{-1}]}{(x^2 + 4x^{-2} - 5)}.$$ Observe that you have the factorization of ideals $$x^2 + 4x^{-2} - 5 = (x-2)(x+2)(x-1)(x+1)$$ in the ring $frac{mathbb{C}[x,x^{-1}]}{(x^2 + 4x^{-2} - 5)}$. These are pairwise coprime maximal ideals, so the Chinese remainder theorem says that $$frac{mathbb{C}[x,x^{-1}]}{(x^2 + 4x^{-2} - 5)}cong frac{mathbb{C}[x,x^{-1}]}{(x-2)}times frac{mathbb{C}[x,x^{-1}]}{(x+2)}times frac{mathbb{C}[x,x^{-1}]}{(x-1)}times frac{mathbb{C}[x,x^{-1}]}{(x+1)}$$$$cong mathbb{C}times mathbb{C}times mathbb{C}times mathbb{C}.$$ Unraveling definitions, we see that $Rcong mathbb{C}^4$, the isomorphism being $$f(x,y)mapsto (f(2,1), f(-2, -1), f(1,2), f(-1,-2)).$$

Now, to understand what the ideals of $R$ are, you need to know what the ideals of $mathbb{C}^4$ are. There are $2^4$ ideals of $mathbb{C}^4$, corresponding to subsets $Ssubseteq{1,2,3,4}$; namely, the ideals $$I_S = {(a_1,a_2,a_3,a_4) : a_i = 0,,,mathrm{ if },,,iin S}.$$ There are therefore $2^4$ ideals of $R$, corresponding to subsets $Ssubseteq{1,2,3,4}$; namely the ideals $$J_S = {f : (f(2,1), f(-2,-1), f(1,2), f(-1,-2))in I_S}.$$ I hope this analysis works!

Here's a procedure that works more generally: the ideal $I := (x^2 + y^2 - 5, xy - 2)$ has height $2$ in $mathbb{C}[x,y]$. To see this, note that $x^2 + y^2 - 5$ (or $xy - 2$) is irreducible over $mathbb{C}$ (e.g. by Eisenstein), hence generates a height $1$ prime ideal in $mathbb{C}[x,y]$, which does not contain the other generator. More generally yet, the generators of $I$ form a regular sequence, and any ideal generated by a regular sequence has height equal to the length of the sequence.

Thus, the quotient $R := mathbb{C}[x,y]/I$ has Krull dimension $le dim mathbb{C}[x,y] - text{ht}(I) = 0$, so $R$ is an Artinian ring. Every Artinian ring is a finite product of Artinian local rings, and ideals in a finite product are products of ideals, so writing $R = prod_{i=1}^n R_i$, the number of $R$-ideals is the product of the numbers of $R_i$-ideals.

In this particular case, $R = R_1 times cdots times R_4$, each $R_i cong mathbb{C}$ by the Nullstellensatz, and $mathbb{C}$ has precisely $2$ ideals, so $R$ has a total of $2^4 = 16$ ideals.