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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
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6. Ohm, J., Semivaluations and groups of divisibility, Can. J. Math. 21 (1969), 576–591.Google Scholar