Exercise 7, § 1

Let $A = \prod_{i=1}^n A_i$, be the product of a finite family of rings, the A, being canonically identified with ideals of A. Let B be a subring of A such that pr_iB = A, for $1 \leq i \leq n$.

(a) Show that, if the A, are Noetherian (resp. Artinian), B is Noetherian (resp. Artinian) (cf. Algebra, Chapter VIII, § 2, Exercise 12).

(b) Let n, be the ideal of B which is the kernel of the restriction of pr, to B. Show that every prime ideal p of B is contained in one of the n_i (use Proposition 1); deduce that, for this index i, pr, p ≠ A,. Show that, if each of the A, contains only a finite number k, of maximal ideals, B contains at most $\sum_{i=1}^n k_i$ maximal ideals.

(c) Let % (A), R(B) be the Jacobson radicals of A and B respectively. Show that R(B) = B ∩ % (A). If each of the A_i contains only a finite number of maximal ideals, show that, for every integer $k > 1$, pr_i((R(B))^k) = (R(A_i))^k for all i and (R(B))^k = (R(A))^k ∩ B (write R(B) as a product of distinct maximal ideals and note that, if m is a maximal ideal of B such that pr_i(m) = m_i is a maximal ideal of A_i, then pr_i(m^k) = m_i^k).