Exercise 26, § 1

(a) Let $A$ be an integral domain such that for all $a \neq 0$ in $A$, the ideal $Aa$ is the intersection of a family $(q_i)$ of saturations of $Aa$ with respect to prime ideals $p_i$ which are minimal among those which contain $a$, such that the domains $A_{p_i}$ are integrally closed (resp. completely integrally closed). Then show that $A$ is integrally closed (resp. completely integrally closed).

(b) Let $A$ be a strongly Laskerian local ring (Chapter IV, § 2, Exercise 28) and $m$ its maximal ideal. Suppose that, for some $a \in A$, $m$ is an immersed prime ideal weakly associated with $A/Aa$ (Chapter IV, § 1, Exercise 17). Show that $Aa : m$ is contained in every ideal which is primary for every prime ideal $p \neq m$ weakly associated with $A/Aa$ (consider the transporter $q : m$ for such a primary ideal $q$, cf. Chapter IV, §2, Exercise 30). If further $A$ is assumed to be an integral domain, show that the relation $p . (Aa : m) = Aa$, for a prime ideal $p$ minimal among those which contain $Aa$, is impossible (if $q$ is the saturation of $A$ with respect to $p$, note that the above relation would imply that $qA_p = qpA_p$ in $A_p$ and conclude with the aid of Exercise 29 (d) of Chapter IV, § 2).

(c) Show that, if $A$ is a strongly Laskerian integral domain and there exists an element $a \# 0$ of $A$ such that there are prime ideals which are weakly associated with $A/Aa$ and immersed, $A$ is not completely integrally closed. (Reduce it to the case considered in (b); in the notation of (b), there then exists $b \in Aa : m$ such that $b \notin Aa$ and $bp \subset am$; deduce by induction on $n$ that $b^n p \subset a^n m$ for all $n$.)

(d) Deduce in particular from (a) and (c) that, for a Noetherian integral domain to be integrally closed, it is necessary and sufficient that for all $a \neq 0$ in $A$ the prime ideals $p$ associated with $A/Aa$ be not immersed and be such that $A_p$ is integrally closed (cf. Chapter VII, § 1, no. 4, Proposition 8).