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Integral Domains which have Finite Character Locally

Published online by Cambridge University Press:  20 November 2018

Kenneth Pacholke
Kenneth Pacholke*
Affiliation:
Northland College, Ashland, Wisconsin 54806
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Abstract

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In recent papers Brewer and Mott have studied integral domains which have finite character globally. This paper concentrates on domains which have finite character locally. Examples include global finite character domains plus Prufer, almost Dedekind, and almost Krull domains. General properties are given, including a valuation-theoretic characterization. The effect of requiring essential and/or rank one valuations is also studied.

Keywords

13F0513G05Finite characterfamily of valuationsalmost Dedekind domainalmost Krull domainPrufer domainessential domaingeneralized Krull domaindomain of finite real characterdomain of Krull type
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 4 , 01 December 1974 , pp. 553 - 558
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Bourbaki, N., Elements de math?matique, Alg?bre commutative, Chapitre 7, Hermann, Paris, 1965.Google Scholar
2. Brewer, J. W., The ideal transform andoverrings of an integral domain, Math. Zeit. 107 (1968), 301-306.Google Scholar
3. Brewer, J. W., Integral domains of finite character, II, J. Reine Angew. Math. 251 (1971), 7-9.Google Scholar
4. Brewer, J. W. and Mott, J. L. Integral domains of finite character, J. Reine Angew. Math. 241 (1970), 34-41.Google Scholar
5. Gilmer, R., Integral domains which are almost Dedekind, Proc. Amer. Math. Soc. 15 (1964), 813-818.Google Scholar
6. Gilmer, R., Multiplicative ideal theory, Queen's Univ., Kingston, Ontario, Canada, 1968.Google Scholar
7. Gilmer, R. and Heinzer, W., Overrings of Prufer domain, II, J. Algebra 7 (1967), 281-302.Google Scholar
8. Gilmer, R. and Heinzer, W., Irredundant intersections of valuation rings, Math. Zeit. 103 (1968), 306-317.Google Scholar
9. Griffin, M., Some results on v-multiplication rings, Canad. J. Math. 19 (1967), 710-722.Google Scholar
10. Griffin, M., Families of finite character and essential valuations, Trans. Amer. Math. Soc. 130 (1968), 75-85.Google Scholar
11. Helmer, O., Divisibility properties of integral functions, Duke Math. J. 6 (1940), 345-356.Google Scholar
12. Henriksen, M., On the prime ideals of the ring of entire functions, Pacific Math. J. 3 (1953), 711-720.Google Scholar
13. Mann, H. B., Introduction to algebraic number theory, Ohio State Univ. Press, Columbus, Ohio, 1955.Google Scholar
14. Nagata, M., A treatise on the 14th problem of Hilbert, Mem. Coll. Sci. Kyoto Univ. 30 (1956), 57-82.Google Scholar
15. Ohm, J., Some counterexamples related to integral closure in D[[x]], Trans. Amer. Math. Soc. 122 (1966), 321-333.Google Scholar
16. Pirtle, E. M. Jr, Integral domains which are almost Krull, J. Sci. Hiroshima Univ. Ser. A-I 32 (1968), 441-447.Google Scholar
17. Ribenboim, P., Anneaux normaux r?els ? carat?refini, Summa Brasil. Math. 3 (1956), 213? 253.Google Scholar
18. Zariski, O. and Samuel, P., Commutative algebra, vol. II, Van Nostrand, Princeton, New Jersey, 1960.Google Scholar