Exercise 17, § 2

Let A be an integral domain.

(a) For $A$ to be Bezoutian ($§ 1$, Exercise 20), it is necessary and sufficient that it be Priiferian and pseudo-Bezoutian ($§ 1$, Exercise 21).

(b) For $A$ to be a Dedekind domain, it is necessary and sufficient that it be a Priiferian Krull domain (show that every divisorial ideal of $A$ is finitely generated, using Exercise 12(x) and $§ 1$, Exercise 11 (a); deduce that every prime ideal $\neq 0$ of $A$ is maximal (Exercise 13(b)), then that every prime ideal of $A$ is finitely generated and conclude with the aid of Chapter II, $§ 1$, Exercise 6).

(c) For $A$ to be a Dedekind domain, it is necessary and sufficient that it be Priiferian and strongly Laskerian (Chapter IV, $§ 2$, Exercise 28). (Observe that, if $A$ is Pruferian and strongly Laskerian, every maximal ideal $m$ of $A$ is such that $A_m$ is a discretevaluation ring (Chapter VI, $§ 3$, Exercise 8), then that for all non-zero $x \in A$ there is a product of a finite number of maximal ideals of $A$ contained in the ideal $Ax$ and conclude that $A$ is a Krull domain.)