Geodesic
An Essential Integral Domain with a Non-Essential Localization
Canadian journal of mathematics, Tome 33 (1981) no. 2, pp. 400-403

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An integral domain D is said to be an essential domain if D is an intersection of valuation rings that are localizations of D. D is called a v-multiplication ring if the finite divisorial ideals of D form a group. Griffin has shown [2, pp. 717-718] that every v-multiplication ring is an essential domain, and that an essential domain 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. It is noted in [4, p. 860] that any localization of a v-multiplication ring is again a v-multiplication ring. In this vein, Joe Mott has asked whether a localization of an essential domain must again be an essential domain. An example of an essential domain that is not a v-multiplication ring is given in [4], however it can be seen for this example that each localization is again an essential domain [6]. 
Heinzer, William. An Essential Integral Domain with a Non-Essential Localization. Canadian journal of mathematics, Tome 33 (1981) no. 2, pp. 400-403. doi: 10.4153/CJM-1981-034-8
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