Exercise 29, § 2

Let A be a strongly Laskerian ring (Exercise 28).

(a) Let a, b be two ideals of A, ab = \bigcap_{i=1}^{n} q_i, a reduced primary decomposition of ab in A and c the intersection of those q_i whose radicals do not contain b; show that a \subset c (if q_i is such that its radicalp, does not contain b and x \in b and x \notin p_i, consider the product ax).

(b) Deduce from (a) that there exist three ideals a', b', c and an integer s > 0 such that a^s \subset a', b^s \subset b', (a + b)^s \subset c and

ab = a' \cap b = a \cap b' = a \cap b \cap c.

(c) Show that for every finite family (a_k) of ideals of A, there exists an integer m > 0 such that \bigcap_k a_k^m \subset \prod_k a_k (apply (b) arguing by induction on the number of ideals a_k).

(d) For every ideal a of A show that the intersection b = \bigcap_{n=1}^{\infty} a^n is the ideal of the x \in A such that x \in xa (apply (b) to the product (Ax)a). Deduce that b is the intersection of the primary components of {0} (for a reduced primary decomposition of {0}) whose radicals meet 1 + a.

(e) Deduce from (d) that, for every prime ideal p of A, the intersection of the symbolic powers p^{(n)} (Exercise 18 (b)) for n \in \mathbf{N} is the ideal of those x \in A such that x \in xp (consider the ring A_p).