Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13C05

Retrieve articles in all journals with MSC: 13C05


Additional Information

Keywords: Bézout ring, quotient ring, residue field, quasi-local ring, Prüfer domain
Article copyright: © Copyright 1975 American Mathematical Society