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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Finite unions of ideals and modules


Authors: Philip Quartararo and H. S. Butts
Journal: Proc. Amer. Math. Soc. 52 (1975), 91-96
MSC: Primary 13C05
DOI: https://doi.org/10.1090/S0002-9939-1975-0382249-5
MathSciNet review: 0382249
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Abstract: We say that a commutative ring $ R$ is a $ u$-ring provided $ R$ has the property that an ideal contained in a finite union of ideals must be contained in one of those ideals; and a $ um$-ring is a ring $ R$ with the property that an $ R$-module which is equal to a finite union of submodules must be equal to one of them. The primary purpose of this paper is to characterize $ u$-rings and $ um$-rings. We show that $ R$ is a $ um$-ring if and only if the residue field $ R/P$ is infinite for each maximal ideal $ P$ of $ R$; and $ R$ is a $ u$-ring if and only if for each maximal ideal $ P$ of $ R$ either the residue field $ R/P$ is infinite or the quotient ring $ {R_p}$ is a Bézout ring.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0382249-5
Keywords: Bézout ring, quotient ring, residue field, quasi-local ring, Prüfer domain
Article copyright: © Copyright 1975 American Mathematical Society