Exercise 21, § 1

An integral domain $A$ is called pseudo-Bezoutian (resp. pseudo-principal) if, in the notation of Exercise 5, the group $K^*/U$ is lattice-ordered (completely lattice ordered). Every Bezoutian \* (resp. factorial) \* domain is pseudo-Bezoutian (resp. pseudo-principal) . Every pseudo-principal domain is pseudo-Bezoutian. A valuation ring whose order group is $\mathbf{R}$ is pseudo-principal but not a principal ideal domain. Every pseudo-Bezoutian (resp. pseudo-principal) domain is integrally closed (resp. completely integrally closed) (use Exercise 6). \ Every Noetherian pseudo-Bezoutian domain is factorial. \* Give an example of an integrally closed Noetherian (and therefore Krull) domain which is not pseudo-Bezoutian (cf. Exercise 2). If $A$ is pseudo-Bezoutian, so is $S^{-1}A$ for every multiplicative subset $S$ of $A$ such that $0 \notin S$ (use Chapter II, § 2, Exercise 1).