Exercise 15, § 4

Let A be a noetherian ring, B an integral A-algebra having only a finite number of minimal prime ideals $p_1, ..., p_n$. For $1 \leq i \leq n$, let us denote by $q_i$ the inverse image of $p_i$ in A and suppose that $\kappa(p_i)$ is a finite extension of $\kappa(q_i)$. Then, for every element x of B, the topological subspace V(x) of Spec(B) has only a finite number of irreducible components. (Reduce to the case where B is integrally closed and contains A, and use Exercise 14.)