Exercise 9, § 1

(a) Let $(\mathfrak{a}_i)_{i=1}^n$ be a finite family of ideals of a ring $A$ which are relatively prime in pairs and such that no $\mathfrak{a}_i$ is of the form $b c = b \cap c$, where $b$ and $c$ are two relatively prime ideals. If $(\mathfrak{a}'_j)_{1 \leq j \leq m}$ is another family of ideals of $A$ which are relatively prime in pairs, such that $a_1 a_2 \ldots a_n = a'_1 a'_2 \ldots a'_m$, show that $m \leq n$; if $r n = n$, there exists a permutation $\pi$ on $[1, n]$ such that $a'_i = a_{\pi(i)}$ for $1 \leq i \leq n$ (use Proposition 5 and Algebra, Chapter VIII, § 1, Exercise 1 (d)).

(b) Let $A$ be a Noetherian ring. Show that every ideal $\mathfrak{a}$ of $A$ is equal to the product of a finite number of ideals which are relatively prime in pairs, none of which is the product of two relatively prime ideals. (Argue by reductio ad absurdurn considering a maximal element of the set of ideals not having this property.)