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An Essential Integral Domain with a Non-Essential Localization

Published online by Cambridge University Press:  20 November 2018

William Heinzer
William Heinzer*
Affiliation:
Purdue University, W. Lafayette, Indiana
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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].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 400 - 403
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Gilmer, R., Multiplicative ideal theory (Marcel Dekker, New York, 1972).Google Scholar
2. Griffin, M., Some results on v-multiplication rings, Can. J. Math. 19 (1967), 710–722.Google Scholar
3. Heinzer, W., Noetherien intersections of integral domains, II, Lecture Notes in Mathematics 311 (1972), 107–119.Google Scholar
4. Heinzer, W. and Ohm, J., An essential ring which is not a v-multiplication ring, Can. J. Math. 21 (1972), 856–861.Google Scholar
5. Heinzer, W. and Ohm, J., Noetherien intersections of integral domains, Trans. Amer. Math. Soc. 167 (1972), 291–308.Google Scholar
6. Mott, J. and Zafrullah, M., On Prufer v-multiplication domains, to appear.Google Scholar