An Essential Ring Which is Not A <i>v</i>-Multiplication Ring

Abstract

An integral domain D is called an essential ring if D = ∩ α V α where the V α 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 { V α } which is of finite character (i.e. every nonzero element of D is a non-unit in at most finitely many V α ) is necessarily a v -multiplication ring; but he conjectures that, in general, there exists an essential ring which is not a v -multiplication ring.