Generalization of a Theorem of Clifford

Abstract

We prove that the multiplicative monoid of principal ideals partially ordered by reverse inclusion, called the divisibility theory, of a Bezout ring R with one minimal prime ideal is a factor of the positive cone of a lattice-ordered abelian group by an appropriate filter if the localization of R at its minimal prime ideal is not a field. This result extends a classical result of Clifford (Am. J. Math. 76:631–646, 1954) saying that the divisibility theory of a valuation ring is a Rees factor of the positive cone of a totally ordered abelian group and suggests to modify Kaplansky’s (later disproved) conjecture (Fuchs and Salce, Mathematical Surveys and Monographs 84, 2001) as to whether a Bezout ring whose localization at every minimal prime ideal has at least three ideals is the factor of an appropriate Bezout domain.