A domain R is called a
pseudo-valuation domain if, whenever a prime ideal P contains the product xy of two
elements of the quotient field of R then x ∈ P or y ∈ P. It is shown that a
pseudo-valuation domain which is not a valuation domain is a quasi-local domain
(R,M) such that V = M−1 is a valuation overring with maximal ideal M. The
authors further show that the nonprincipal divisorial ideals of R coincide with the
nonzero ideals of V . These ideas are then applied to the case of a Noetherian
pseudo-valuation domain R. Such a domain R is shown to have all its nonzero
ideals divisorial if and only if each ideal is two-generated. Examples include
valuation rings, certain D + M constructions, and certain rings of algebraic
integers.
