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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

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