An Essential Ring Which is Not A v-Multiplication Ring

Abstract

An integral domain D is called an essential ring if D = ∩αVα where the Vα are valuation rings which are quotient rings of D. D is called a v-multiplication ring if the finite divisorial ideals of D form a group. Griffin [2, pp. 717-718] has observed that every v-multiplication ring is essential and that an essential ring having a defining family of valuation rings {Vα} which is of finite character (i.e. every nonzero element of D is a non-unit in at most finitely many Vα) is necessarily a v-multiplication ring; but he conjectures that, in general, there exists an essential ring which is not a v-multiplication ring.