Vol. 75, No. 1, 1978

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Pseudo-valuation domains

John R. Hedstrom and Evan Green Houston, Jr.

Vol. 75 (1978), No. 1, 137–147

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 = M1 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.

Mathematical Subject Classification 2000
Primary: 13A15
Secondary: 13G05
Received: 11 October 1976
Revised: 18 March 1977
Published: 1 March 1978
John R. Hedstrom
Evan Green Houston, Jr.