Krull modules and completely integrally closed modules

Let [Formula: see text] be a torsion-free module over an integral domain [Formula: see text] with quotient field [Formula: see text]. We define a concept of completely integrally closed modules in order to study Krull modules. It is shown that a Krull module [Formula: see text] is a [Formula: see text]-multiplication module if and only if [Formula: see text] is a maximal [Formula: see text]-submodule and [Formula: see text] for every minimal prime ideal [Formula: see text] of [Formula: see text]. If [Formula: see text] is a finitely generated Krull module, then [Formula: see text] is a Krull module and [Formula: see text]-multiplication module. It is also shown that the following three conditions are equivalent: [Formula: see text] is completely integrally closed, [Formula: see text] is completely integrally closed, and [Formula: see text] is completely integrally closed.

Arithmetic modules over generalized Dedekind domains

Let [Formula: see text] be a finitely generated torsion-free module over a generalized Dedekind domain [Formula: see text]. It is shown that if [Formula: see text] is a projective [Formula: see text]-module, then it is a generalized Dedekind module and [Formula: see text]-multiplication module. In case [Formula: see text] is Noetherian it is shown that [Formula: see text] is either a generalized Dedekind module or a Krull module. Furthermore, the polynomial module [Formula: see text] is a generalized Dedekind [Formula: see text]-module (a Krull [Formula: see text]-module) if [Formula: see text] is a generalized Dedekind module (a Krull module), respectively.

A Study of The Density Property in Module Theory

In this paper, there are two main objectives. The first objective is to study the relationship between the density property and some modules in detail, for instance; semisimple and divisible modules. The Addition complement has a good relationship with the density property of the modules as this importance is highlighted by any submodule N of M has an addition complement with Rad(M)=0. The second objective is to clarify the relationship between the density property and the essential submodules with some examples. As an example of this relationship, we studied the torsion-free module and its relationship with the essential submodules in module M.

ON LCM-STABLE MODULES

A torsion-free module M over a commutative integral domain R is said to be LCM-stable over R if (Ra ∩ Rb)M = Ma ∩ Mb for all a, b ∈ R. We show that if the module M is LCM-stable over a GCD-domain R, then the polynomial module M[X] is LCM-stable over R[X]; if R is a w-coherent locally GCD-domain, then LCM-stability and reflexivity are equivalent for w-finite type torsion-free R-modules. Finally, we introduce the concept of w-LCM-stability for modules over a domain. Then we characterize when the module M is w-LCM-stable over the domain in terms of localizations and t-Nagata modules, respectively. Also we characterize Prüfer v-multiplication domains and Krull domains in terms of w-LCM-stability.

On the Unions of Ascending Chains of Direct Sums of Ideals of h-Local Prüfer Domains

In this work, we investigate conditions under which unions of ascending chains of modules which are isomorphic to direct sums of ideals of an integral domain are again isomorphic to direct sums of ideals. We obtain generalizations of the Pontryagin-Hill theorems for modules which are direct sums of ideals of h-local Prüfer domains. Particularly, we prove that a torsion-free module over a Dedekind domain with a countable number of maximal ideals is isomorphic to a direct sum of ideals if it is the union of a countable ascending chain of pure submodules which are isomorphic to direct sums of ideals.

On Simple Rings with Maximal Annihilator Right Ideals

If R is a simple ring with 1 which contains a maximal annihilator right ideal then R is the endomorphism ring of a unital torsion-free module over an integral domain.We first prove the following:Let R be a ring with 1. If a ϵ R such that (a)r = {r ϵ R|ar = 0} is a maximal annihilator right ideal then HomR(aR, aR) is an integral domain.