Exercise 29, § 2

Let $A$ be a strongly Laskerian ring (exerc. 28).

a) Let a, b be two ideals of $A$, $ab = \bigcap_{i=1}^{n} q_i$ a reduced primary decomposition of $ab$ in $A$, c the intersection of those of the $q_i$ whose root does not contain b; prove that one has $a \subset c$ (if $q_i$ is such that its root $p_i$ does not contain b and if $x \in b$ 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) Prove 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) by reasoning by induction on the number of ideals a_k.

d) For every ideal a of $A$, prove that the intersection b = $\bigcap_{n=1}^{\infty} a^n$ is the ideal of the x ∈ 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 roots meet $1 + a$.

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