Pseudo-valuation domains

Abstract

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 1? then JC E P or y G 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~] 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.