calculating minimal prime ideals Is there a "general approach" to determine the minimal prime ideals over an ideal $J$?
I checked some books and didn't find a general approach. Maybe the theory of Gröbner bases is related to these type of questions, but I think this theory is beyond the scope of my introductory course.
The original exercise, which I had to solve, asked me to find the irreducible components, with respect to the Zariski topology, of the algebraic set $S=Z(x^2+y^2 -1, x^2 - z^2 -1)$ in $\mathbb{C}^3$. 
I proposed that this question is equivalent to finding the minimal prime ideals of $J=(x^2+y^2 -1, x^2 - z^2 -1)$. This led me to the question whether there was a general approach for finding the minimal prime ideals over an ideal $I$? 
It would be convenient if someone could illustrate me the general techinque, so I can imitate it for other cases also. Thanks in advance.
 A: The minimal prime ideals of $A$ are precisely the generic points of $\mathrm{Spec}(A)$. Here, a point of a topological space $\eta \in X$ is called generic if $\eta \prec x \Rightarrow \eta = x$ with respect to the specialization preorder, i.e. $\eta$ has no proper generizations. In a soper space, generic points correspond 1:1 to the irreducible components via $\eta \mapsto \overline{\{\eta\}}$.
In your example, $S$ is the regular intersection of a cylinder with a hyperbolic surface, resulting in two circles. In fact, observe
$$\begin{eqnarray*}
S&=&V(x^2+y^2-1,z^2+y^2)\\
&=&V(x^2+y^2-1,(z+iy)(z-iy))\\
&=&V(x^2+y^2-1,z+iy) \cup V(x^2+y^2-1,z-iy).
\end{eqnarray*}$$
Both of these algebraic sets $\subseteq \mathbb{A}^3$ are isomorphic to the circle $\subseteq \mathbb{A}^2$. Hence these are the irreducible components.
You can also do this algebraically (but this is more complicated than the geometric approach, in my opinion) i.e. with your approach using minimal prime ideals. We have
$$\begin{eqnarray*}
\Gamma(S)&=&k[x,y]/(x^2+y^2-1)[z]/((z+iy)(z-iy)) \\
&\stackrel{\mathrm{CRT}}{\cong}& k[x,y]/(x^2+y^2-1)[z]/(z+iy) \times k[x,y]/(x^2+y^2-1)[z]/(z-iy) \\
&\cong & k[x,y]/(x^2+y^2-1) \times k[x,y]/(x^2+y^2-1)
\end{eqnarray*}$$
This is a product of two (isomorphic) integral domains. Hence there are two minimal prime ideals, namely $\langle (1,0) \rangle$ and $\langle (0,1) \rangle$.
