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Integral ∨-ideals

Published online by Cambridge University Press:  18 May 2009

David F. Anderson
David F. Anderson
Affiliation:
Department of Mathematics, The University of Tennessee, Knoxvelle, Tennessee 37916
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Let R be an integral domain with quotient field K. A fractional ideal I of R is a ∨-ideal if I is the intersection of all the principal fractional ideals of R which contain I. If I is an integral ∨-ideal, at first one is tempted to think that I is actually just the intersection of the principal integral ideals which contain I.However, this is not true. For example, if R is a Dedekind domain, then all integral ideals are ∨-ideals. Thus a maximal ideal of R is an intersection of principal integral ideals if and only if it is actually principal. Hence, if R is a Dedekind domain, each integral ∨-ideal is an intersection of principal integral ideals precisely when R is a PID.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 22 , Issue 2 , July 1981 , pp. 167 - 172
Copyright
Copyright © Glasgow Mathematical Journal Trust 1981

References

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