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Defining Families for Integral Domains of Real Finite Character

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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Throughout this paper R and D will denote integral domains with the same quotient field K. A set of integral domains {Di} i∊I with quotient field K will be said to have FC (“finite character” or “finiteness condition“) if 0 ≠ ξ ∊ K implies ξ is a unit of Di for all but finitely many i. If ∩i∊IDi also has quotient field K, then {Di} has FC if and only if every non-zero element in ∩i∊IDi is a non-unit in at most finitely many Di. A non-empty set {Vi}i∊:I of rank one valuation rings with quotient field K will be called a defining family of real R-representativesfor D if {Vi} i∊:I has FC, R (⊄ ∩i∊IVi, and D = R∩ (∩i∊I Vi).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 6 , December 1972 , pp. 1170 - 1177
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Bourbaki, N., Algebre commutative, Chapters 5 and 6 (Hermann, Paris, 1964).Google Scholar
2. Brewer, J. and Mott, J., Integral domains of finite character, J. Reine Angew. Math. 241 (1970), 34–41.Google Scholar
3. Griffin, M., Families of finite character and essential valuations, Trans. Amer. Math. Soc. 130 (1968), 75–85.Google Scholar
4. Heinzer, W. and Ohm, J., Noetherian intersections of integral domains, Trans. Amer. Math. Soc. 167 (1972), 291–308.Google Scholar
5. Krull, W., Beiträge zur Arithmetik kommutativer Integritatsbereiche, Math. Z. 41 (1936), 545–577.Google Scholar
Nagata, M., Local rings (Interscience, New York, 1962).Google Scholar
Ohm, J., Some counterexamples related to integral closure in D[[X]], Trans. Amer. Math. Soc. 122 (1966), 321-333Google Scholar
Pontrjagin, L., Topological groups (Princeton Univ. Press, Princeton, 1939).Google Scholar
Ribenboim, P., Anneaux normaux réels à caractère fini, Summa Brasil. Math. 3 (1956), 213-253.Google Scholar