Exercise 13, § 2

(a) In a Priiferian domain A, let $a, b$ be two finitely generated ideals; show that, if $b \notin a$, there exists a finitely generated ideal $c$ such that $a \subset c$, $a \neq c$ and $bc \subset a$ (consider the ideal $a + b$).

(b) Deduce from (a) that, if $a$ is a finitely generated ideal in a Priiferian domain A, then in the ring $A/a$ every element which is not a divisor of zero is invertible. In particular, a prime ideal of A cannot be finitely generated if it is maximal.

(c) Show that in a Priiferian domain A two prime ideals $p, p'$ neither of which is contained in the other are relatively prime (consider, for every maximal ideal $m$ of A, the ideal $pA_m + p'A_m$).

(d) In a Priiferian domain A let $a$ be a finitely generated ideal. For $a$ to be primal (Chapter IV, § 2, Exercise 33), it is necessary and sufficient that $a$ be contained only in a single maximal ideal of A (use (b)); $a$ is then irreducible and quasi-prime (Chapter IV, § 2, Exercise 34), so that, for finitely generated ideals, these three notions coincide. For $a$ to be primary, it is necessary and sufficient that it be contained only in a single prime ideal (necessarily maximal) $m$; for $a$ to be strongly primary (Chapter IV, § 2, Exercise 27), it is necessary and sufficient also that $mA_m$ be principal.