Finite unions of ideals and modules

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.