Exercise 26, § 2

Let $A$ be a Laskerian ring in which every ideal $\#A$ has a unique reduced primary decomposition in $A$.

(a) Show that for every ideal $a \neq A$ all the elements of $\mathrm{Ass}_f(A/a)$ are minimal in this set (use Exercise 25 (b)). Let $\{0\} = \bigcap_{1 \leq i \leq n} q_i$ be a reduced primary decomposition in $A$ and let $p_i$ be the radical of $q_i$ ($1 \leq i \leq n$). Show that every prime ideal of $A$ distinct from the $p_i$ is maximal (use Exercise 24 (c)). If $p$ is a prime ideal of $A$, $q$ an ideal contained in $p$ and such that $p/q$ is a nil ideal in $A/q$, show that every ideal $a$ such that $q \subset a \subset p$ is $p$-primary. Deduce that if one of the $p_i$ is not maximal, $p_i$ is the only $p_i$-primary ideal and $p_i^2 = p_i$ (use Exercise 16).

(b) Show that in $A$ every increasing sequence of ideals equal to their radicals is stationary. Deduce that every strictly increasing sequence $(a_i)$ of ideals such that every quotient $a_n/a_{n-1}$ contains non-nilpotent elements in $A/a_{n-1}$ is finite. Conclude from this that for every prime ideal $p$ of $A$ there exists a primary ideal with radical $p$ which is finitely generated. In particular, each of the $p_i$ which is not maximal is finitely generated and therefore contains an idempotent $e_i$ which generates it (Chapter 11, § 4, Exercise 15).

(c) Show that, for $i \neq j$ $p_i + p_j = A$ (consider $e_i + e_j - e_i e_j$ if $p_i$ and $p_j$ are not maximal). Conclude that $A$ is isomorphic to the product of a finite number of rings $A_i = A/q_i$ such that: either $A$, is a local ring whose maximal ideal is a nil ideal; or $A$ is an integral domain in which every prime ideal $\neq \{0\}$ is maximal and in which every ideal $\neq \{0\}$ is only contained in a finite number of maximal ideals (cf. § 1, Exercise 2). \* Obtain the converse.