Weakly distributive modules. Applications to supplement submodules

In this paper, we define and study weakly distributive modules as a proper generalization of distributive modules.We prove that, weakly distributive supplemented modules are amply supplemented. In a weakly distributive supplemented module every submodule has a unique coclosure. This generalizes a result of Ganesan and Vanaja. We prove that π-projective duo modules, in particular commutative rings, are weakly distributive. Using this result we obtain that in a commutative ring supplements are unique. This generalizes a result of Camillo and Lima. We also prove that any weakly distributive ⊕-supplemented module is quasi-discrete.     

Generating dual Baer modules via fully invariant submodules

Abstract

In this paper, we introduce a concept of a dual F-Baer module M where F is the fully invariant submodule of M, by this means we deal with generating dual Baer modules. We investigate direct sums of dual F -Baer modules M by exerting the notion of relatively dual F-Baer modules. We also obtain applications of dual F-Baer modules to rings and the preradical Z*(·).     

Singular and nonsingular modules relative to a torsion theory

In this paper, we introduce and study torsion-theoretic generalizations of singular and nonsingular modules by using the concept of τ-essential submodule for a hereditary torsion theory τ. We introduce two new module classes called τ-singular and non-τ-singular modules. We investigate some properties of these module classes and present some examples to show that these new module classes are different from singular and nonsingular modules. We give a characterization of τ-semisimple rings via non-τ-singular modules. We prove that if M∕τ(M) is non-τ-singular for a module M, then every submodule of M has a unique τ-closure. We give some properties of the torsion theory generated by the class of all τ-singular modules. We obtain a decomposition theorem for a strongly τ-extending module by using non-τ-singular modules.     

Copolyform Modules

Abstract

In this note all rings R are associative with identity, all modules are unitary right modules and we denote the category of all such R-modules by Mod-R. Let M ? Mod-R and A ? M. Corational submodules of A in M are defined and studied. Examples of modules M for which every submodule has a smallest corational submodule in M and an example of a module M with a submodule A which has no minimal corational submodule in M are given. In the second half of the paper copolyform modules are defined and we characterize when a finite direct sum of weakly supplemented copolyform modules is copolyform.     

PRIME AND RADICAL SUBMODULES OF MODULES OVER COMMUTATIVE RINGS

ABSTRACT

In this paper, unless otherwise stated, all rings are commutative with identity and all modules are unital. We give sufficient conditions to ensure that a submodule has a module-reduced primary decomposition. In general, the radical of a primary submodule is not prime and the radical does not split intersections of submodules, as is valid in the ideal case. We study sufficient conditions for which these properties hold in the module setting. These conditions involve dimension arguments, consideration of finitely generated modules, and the spectrum of a given prime ideal. Further, we consider the computational problem of finding a Gröbner basis of both the colon and the radical of a submodule. A characterization of the elements of the colon is given, along with a method of computing the radical of a submodule in certain cases.     

Strongly radical supplemented modules

Z?schinger studied modules whose radicals have supplements and called these modules radical supplemented. Motivated by this, we call a module strongly radical supplemented (briefly srs) if every submodule containing the radical has a supplement. We prove that every (finitely generated) left module is an srs-module if and only if the ring is left (semi)perfect. Over a local Dedekind domain, srs-modules and radical supplemented modules coincide. Over a nonlocal Dedekind domain, an srs-module is the sum of its torsion submodule and the radical submodule.     

On Fully Idempotent Modules

A submodule N of a module M is idempotent if N = Hom(M, N)N. The module M is fully idempotent if every submodule of M is idempotent. We prove that over a commutative ring, cyclic idempotent submodules of any module are direct summands. Counterexamples are given to show that this result is not true in general. It is shown that over commutative Noetherian rings, the fully idempotent modules are precisely the semisimple modules. We also show that the commutative rings over which every module is fully idempotent are exactly the semisimple rings. Idempotent submodules of free modules are characterized.     

On Cofinitely Lifting and Cofinitely Weak Lifting Modules

We say that a module M is lifting if M is amply supplemented and every supplement submodule of M is a direct summand. The module M is called cofinitely lifting if it is amply cofinitely supplemented and every supplement of any cofinite submodule of M is a direct summand. In this article various properties of cofinitely lifting modules are given. In addition, a generalization of cofinitely lifting modules is investigated.     

On the Countability of Noetherian Dimension of Modules

Abstract

It is shown that a module M has countable Noetherian dimension if and only if the lengths of ascending chains of submodules of M has a countable upper bound. This shows in particular that every submodule of a module with countable Noetherian dimension is countably generated. It is proved that modules with Noetherian dimension over locally Noetherian rings have countable Noetherian dimension. We also observe that ω^ω is a universal upper bound for the lengths of all chains in Artinian modules over commutative rings.     

Another type of dimensions of modules and rings

Abstract

We say that a class Q of left R-modules is a monic class if a nonzero submodule of a module in Q is also a module in Q. For a monic class Q, we define a Q-dimension of modules that measures how far modules are from the modules in Q. For a monic class Q of indecomposable modules we characterize rings whose modules have Q-dimension. We prove that for an artinian principal ideal ring the Q-dimension coincides with the uniserial dimension. We also characterize when every module has Q-dimension.     

Infinite Direct Sums of Lifting Modules

A module M over a ring R is called a lifting module if every submodule A of M contains a direct summand K of M such that A/K is a small submodule of M/K (e.g., local modules are lifting). It is known that a (finite) direct sum of lifting modules need not be lifting. We prove that R is right Noetherian and indecomposable injective right R-modules are hollow if and only if every injective right R-module is a direct sum of lifting modules. We also discuss the case when an infinite direct sum of finitely generated modules containing its radical as a small submodule is lifting.     

Weakly dual automorphism invariant and superfluous cotightness

Abstract

The aim of the present paper is to introduce and study the dual concepts of weakly automorphism invariant modules and essential tightness. These notions are non-trivial generalizations of both weakly projectivity, dual automorphism invariant property and cotightness. We obtain certain relations between weakly projective modules, weakly dual automorphism invariant modules and superfluous cotight modules. It is proved that: (1) for right perfect rings, every module is a direct summand of a weakly dual automorphism invariant module and (2) weakly dual automorphism invariant modules are precisely superfluous cotight modules.     

Quasi Co-Hopfian Modules and Applications

We carry out an extensive study of modules M _R with the property that M/f(M) is singular for all injective endomorphisms f of M. Such modules called “quasi co-Hopfian”, generalize co-Hopfian modules. It is shown that a ring R is semisimple if and only if every quasi co-Hopfian R-module is co-Hopfian. Every module contains a unique largest fully invariant quasi co-Hopfian submodule. This submodule is determined for some modules including the semisimple ones. Over right nonsingular rings several equivalent conditions to being quasi co-Hopfian are given. Modules with all submodules quasi co-Hopfian are called “completely quasi co-Hopfian” (cqcH). Over right nonsingular rings and over certain right Noetherian rings, it is proved that every finite reduced rank module is cqcH. For a right nonsingular ring which is right semi-Artinian (resp. right FBN) the class of cqcH modules is the same as the class of finite reduced rank modules if and only if there are only finitely many isomorphism classes of nonsingular R-modules which are simple (resp. indecomposable injective).     

Weakly regular modules over normal rings

Under study are some conditions for the weakly regular modules to be closed under direct sums and the rings over which all modules are weakly regular. For an arbitrary right R-module M, we prove that every module in the category σ(M) is weakly regular if and only if each module in σ(M) is either semisimple or contains a nonzero M-injective submodule. We describe the normal rings over which all modules are weakly regular.     

MODULES WITH FULLY INVARIANT SUBMODULES ESSENTIAL IN FULLY INVARIANT SUMMANDS

ABSTRACT

A module M is called (strongly) FI-extending if every fully invariant submodule is essential in a (fully invariant) direct summand. The class of strongly FI-extending modules is properly contained in the class of FI-extending modules and includes all nonsingular FI-extending (hence nonsingular extending) modules and all semiprime FI-exten ding rings. In this paper we examine the behavior of the class of strongly FI-extending modules with respect to the preservation of this property in submodules, direct summands, direct sums, and endomorphism rings.     

Properties of <Emphasis Type="Italic">P</Emphasis>-coherent and Baer modules

M is called a P-coherent (resp. PP) module if its every principal submodule is finitely presented (resp. projective). M is said to be a Baer module if the annihilator of its every subset is a direct summand of R. In this paper, we investigate the properties of P-coherent, PP and Baer modules. Some known results are extended.     

h-Divisible Modules

Abstract

We investigate classes of h-divisible modules over domains and a class of domains over which every module has a divisible envelope.     

Modules in Which Every Fully Invariant Submodule is Essential in a Direct Summand

A module M is called extending if every submodule of M is essential in a direct summand. We call a module FI-extending if every fully invariant submodule is essential in a direct summand. Initially we develop basic properties in the general module setting. For example, in contrast to extending modules, a direct sum of FI-extending modules is FI-extending. Later we largely focus on the specific case when a ring is FI-extending (considered as a module over itself). Again, unlike the extending property, the FI-extending property is shown to carry over to matrix rings. Several results on ring direct decompositions of FI-extending rings are obtained, including a proper generalization of a result of C. Faith on the splitting-off of the maximal regular ideal in a continuous ring.