GENERALIZATION OF A THEOREM OF CLIFFORD

Abstract. The multiplicative monoid of principal ideals partially ordered by reverse in-clusion, 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 [6] 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 [8] as to a Bezout ring whose localization at every minimal prime ideal is not a field, is the factor of an appropriate Bezout domain. 1