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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Finite unions of ideals and modules
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by Philip Quartararo and H. S. Butts PDF
Proc. Amer. Math. Soc. 52 (1975), 91-96 Request permission

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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Additional Information
  • © Copyright 1975 American Mathematical Society
  • 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