A note on modules (original) (raw)

Few remarks on essential modules

arXiv (Cornell University), 2022

In the present paper, modules over integral domains and principal ideal domains that are proper essential extensions of some submodules are classified. We introduce a new class of modules that we call SM modules and show that the class of Artinian modules, locally finite modules, and modules of finite lengths are all proper subclasses of SM modules. We also show that non-semisimple SM modules possess essential socles. Further, we show that non-semieimple modules over integral domains with nonempty torsion-free parts do not possess essential socles.

A generalization of supplemented modules

Algebra and Discrete Mathematics

Let R be an arbitrary ring with identity and M a right R-module. In this paper, we introduce a class of modules which is an analogous of δ-supplemented modules defined by Kosan. The module M is called principally δ-supplemented, for all m ∈ M there exists a submodule A of M with M = mR+A and (mR)∩A δsmall in A. We prove that some results of δ-supplemented modules can be extended to principally δ-supplemented modules for this general settings. We supply some examples showing that there are principally δ-supplemented modules but not δ-supplemented. We also introduce principally δ-semiperfect modules as a generalization of δ-semiperfect modules and investigate their properties.

A Note on⊕-Cofinitely Supplemented Modules

2007

Let R be a ring and M a right R-module. In this note, we show that a quotient of an⊕-cofinitely supplemented module is not in general⊕-cofinitely supplemented and prove that if a module M is an⊕-cofinitely supplemented multiplication module with Rad (M)≪ M, then M can be written as an irredundant sum of local direct summand of M.

On Μ-Singular and Μ-Extending Modules

2012

Let M be a module and µ be a class of modules in Mod R which is closed under isomorphisms and submodules. As a generalization of essential submodules Ozcan in (8) defines a µ-essential submodule provided it has a non-zero intersection with any non-zero submodule in µ. We define and investigate µ-singular modules. We also introduce µ-extending and weakly µ-extending modules and mainly study weakly µ-extending modules. We give some characterizations of µ-co-H-rings by weakly µ-extending modules. Let R be a right non-µ-singular ring such that all injective modules are non-µ-singular, then R is right µ-co-H-ring if and only if R is a QF-ring.

Applications of reduced and coreduced modules II

arXiv (Cornell University), 2023

This is the second in a series of papers highlighting the applications of reduced and coreduced modules. Let R be a commutative unital ring and I be an ideal of R. We give the necessary and sufficient conditions in terms of I-reduced and I-coreduced R-modules for the functors Hom R (R/I, −) and Γ I , the I-torsion functor, on the abelian full subcategories of the category of all R-modules to be radicals. These conditions: 1) subsume and unify many results which were proved on a case-by-case basis, 2) provide a setting for the generalisation of Jans' correspondence of an idempotent ideal of a ring with a torsion-torsionfree class, 3) provide answers to open questions that were posed by Rohrer, and 4) lead to a new radical class of rings.

On a Class of ⊕-Supplemented Modules

Contemporary Mathematics, 2014

Let R be an arbitrary ring with identity and M a right R-module. In this paper, we introduce a class of modules which is an analogous to δsupplemented modules and principally ⊕-supplemented modules. The module M is called principally ⊕-δ-supplemented if for any m ∈ M there exists a direct summand A of M such that M = mR + A and mR ∩ A is δ-small in A. We prove that some results of principally ⊕-supplemented modules can be extended to principally ⊕-δ-supplemented modules for this general settings. Several properties of these modules are given and it is shown that the class of principally ⊕-δ-supplemented modules lies strictly between classes of principally ⊕-supplemented modules and principally δ-supplemented modules. We investigate conditions which ensure that any factor modules, direct summands and direct sums of principally ⊕-δ-supplemented modules are also principally ⊕-δ-supplemented. We give a characterization of principally ⊕-δ-supplemented modules over a semisimple ring and a new characterization of principally δsemiperfect rings is obtained by using principally ⊕-δ-supplemented modules.

A note on modules over regular rings

Bulletin of the Australian Mathematical Society

It is shown that a von Neumann regular ring R is left seif-injective if and only if every finitely generated torsion-free left R-module is projective. It is further shown that a countable self-injective strongly regular ring is Artin semi-simple.

Some Results on Weak Essential Submodules

Throughout this paper R represents commutative ring with identity and M is a unitary left R-module. The purpose of this paper is to investigate some new results (up to our knowledge) on the concept of weak essential submodules which introduced by Muna A. Ahmed, where a submodule N of an R-module M is called weak essential, if N ∩ P ≠ (0) for each nonzero semiprime submodule P of M. In this paper we rewrite this definition in another formula. Some new definitions are introduced and various properties of weak essential submodules are considered.