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An Essential Ring Which is Not A v-Multiplication Ring

Published online by Cambridge University Press:  20 November 2018

William Heinzer  and
Jack Ohm
William Heinzer
Affiliation:
Purdue University, Lafayette, Indiana
Jack Ohm
Affiliation:
Louisiana State University, Baton Rouge, Louisiana
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 4 , August 1973 , pp. 856 - 861
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Bourbaki, N., Algebre commutative, Chapters 5, 6, 7 (Hermann, Paris, 1964/65).Google Scholar
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4. Heinzer, W. and Ohm, J., Noetherian intersections of integral domains, Trans. Amer. Math. Soc. 167 (1972), 291–308.Google Scholar
5. Jaffard, P., Les systèmes d'idéaux (Dunod, Paris, 1960).Google Scholar
6. Ohm, J., Semivaluations and groups of divisibility, Can. J. Math. 21 (1969), 576–591.Google Scholar