Small Submodules Research Articles

Some Properties of S-Semiannihilator Small Submodules and S-Small Submodules with respect to a Submodule

Let R be a commutative ring with nonzero identity, S⊆R be a multiplicatively closed subset of R, and M be a unital R-module. In this article, we introduce the concepts of S-semiannihilator small submodules and S-T-small submodules as generalizations of S-small submodules. We investigate some basic properties of them and give some characterizations of such submodules, especially for (finitely generated faithful) multiplication modules.

Study on-purely small sub-modules

D is a unitary left and R is an associative ring with identity. Module R. We call a submodule C of an R-module D called purely-small (for short Pu-small), denoted by (C<<Pu D) if C + B ≠ D For every proper pure sub-module B of D. When D is finitely generated multiplication and faithful, we study A purely-small in D in this paper. Conditions when[A:D] purely-small in R are given. This paper determines to investigate some new results on the notion of purely small sub-modules introduced by Ahmad, Dhari, and Atiya various properties of this kind of sub-modules considered.

A new approach to multiplication modules via (delta)-small submodules

"Let R be a commutative ring and M an R-module. In this work we introduce two new generalizations of multiplication modules via delta-small submodules and small submodules of a fixed module. A module M is said to be (delta)-small multiplication provided for every (delta-)small submodule of N of M, there is an ideal I of R such that N=IM. We study some general properties of both delta-small multiplication modules and also small multiplication modules. A counterexample is presented to state this fact that the class of all delta-small multiplication modules lies exactly between the class of multiplication modules and small multiplication modules. We show that any direct summand of a (delta)-small multiplication module inherits the property."

Large-maximal small submodules

In this study, the ideas of Large-maximal small submodules and Large-maximal small radical of modules are presented. The primary attributes and features of the concept of large-maximal small submodules are provided. Additionally, we discuss the connections between this idea and various submodule kinds with the help of examples and observations that are relevant to our work. Where a proper submodule A of a T-module G is said to be Large-maximal small submodule, if A + B = G where B is a proper submodule of G, then B is Large-Maximal submodule of G. A Large-maximal small radical of module is the sum of all Large-maximal small submodule in G. It will be studied how this concept of radicality relates to other radical conceptions.

Purely Small Submodules and Purely Hollow Modules

The main goal of this paper is to give a new generalizations for two important classes in the category of modules, namely the class of small submodules and the class of hollow modules. They are purely small submodules and purely hollow modules respectively. Various properties of these classes of modules are investigated. The relationship between purely small submodules and P-small submodules which is introduced by Hadi and Ibrahim, is studied. Moreover, another characterization of purely hollow modules is considered.

F-Hollow modules

In previous research, a concept was presented f –small submodules A submodule L of an R-module W is named f –small if L+X=W and W/X are small-singular, then X=W. In this work, we present a new concept, namely f –hollow modules, an R-module W is f-hollow if every submodule in W is an f-small submodule. This concept is an expansion of Hollow modules. Some characteristics are given in expansion to their relationship to a few of the concepts and the comparability between them is examined.

Modules with finitely many small submodules

O. A. S. Karamzadeh et al. [On rings with a unique proper essential right ideal, Fund. Math. 183 (2004) 229–244] introduced a class of modules with unique proper essential submodule, such modules are called as ue-modules. A. Azarang and F. Shahrisvand [Rings with only finitely many essential right ideals, Comm. Algebra 47 (2019) 2843–2854] introduced the idea of fe-module and called a module with finitely many essential submodules as a fe-module. Here, we introduce and study the classes of modules dual to them. The class of modules with finitely many small submodules is defined as fs-modules and modules with a unique nonzero small submodule are defined as us-modules.

Pythagorean Fuzzy Small Submodules

In this paper, we introduce the notion of a pythagorean fuzzy small submodule. We prove various characterisations for pythagorean fuzzy small submodules. We provide a relation between a pythagorean fuzzy small submodule and a basic small submodule. In addition, some important properties regarding pythagorean fuzzy small submodules are investigated.

Semi -T- Small Submodules

Let be a ring with identity and be a submodule of a left - module . A submodule of is called - small in denoted by , in case for any submodule of , implies . Submodule of is called semi -T- small in , denoted by , provided for submodule of , implies that . We studied this concept which is a generalization of the small submodules and obtained some related results

L-Hollow modules

To consider R is a commutative ring with unity, be a nonzero unitary left R-module, is known hollow module if each proper submodule of is small. L-hollow module is a strong form of hollow module, where an R-module is known L-hollow module if has a unique maximal submodule which contains each small submodules. The current study deals with this class of modules and give several fundamental properties related with this concept.

On Jacobson – Small Submodules

Let R be an associative ring with identity and let M be a unitary left R–module. As a generalization of small submodule , we introduce Jacobson–small submodule (briefly J–small submodule ) . We state the main properties of J–small submodules and supplying examples and remarks for this concept . Several properties of these submodules are given . Also we introduce Jacobson–hollow modules ( briefly J–hollow ) . We give a characterization of J–hollow modules and gives conditions under which the direct sum of J–hollow modules is J–hollow . We define J–supplemented modules and some types of modules that are related to J–supplemented modules and introduce properties of this types of modules . Also we discuss the relation between them with examples and remarks are needed in our work.

Intuitionistic Fuzzy Small Submodules and Their Properties

The concept of an intuitionistic fuzzy set, which is a generalisation of fuzzy set, was introduced by K. T. Atanassov in 1986. In this paper using of intuitionistic fuzzy small submodules we get some results about this kind of fuzzy submodules. First we give some preliminary properties of intuitionistic fuzzy submodules. Then we attempt to investigate various properties of intuitionistic fuzzy small submodules. A necessary and sufficient condition for intuitionistic fuzzy small submodules is established. We investigate the nature of intuitionistic fuzzy small submodules under direct sum. Also we study on relation between intuitionistic fuzzy small submodules and level subsets of them, and get some interesting results in this sense.

Decompositions of Dual Automorphism Invariant Modules over Semiperfect Rings

A module M is called dual automorphism invariant if whenever X1 and X2 are small submodules of M, then each epimorphism f : M/X1 → M/X2 lifts to an endomorphism g of M. A module M is said to be d-square free (dual square free) if whenever some factor module of M is isomorphic to N2 for a module N then N = 0. We show that each dual automorphism invariant module over a semiperfect ring which is a small epimorphic image of a projective lifting module is a direct sum of cyclic indecomposable d-square free modules. Moreover, we prove that for each module M over a semiperfect ring which is a small epimorphic image of a projective lifting module (e.g., M is a finitely generated module), M is dual automorphism invariant iff M is pseudoprojective. Also, we give the necessary and sufficient conditions for a dual automorphism invariant module over a right perfect ring to be quasiprojective.

Pre-Assessment of Radiated Fields from Small Electronic Submodules

Pre-Assessment of Radiated Fields from Small Electronic Submodules

On the small Krull dimension of modules

ABSTRACTWe introduce and study the concept of small Krull dimension of a module which is Krull-like dimension extension of the concept of DCC on small submodules. Using this concept we extend some of the basic results for modules with this dimension, which are almost similar to the basic properties of modules with Krull dimension. When for a module A with small Krull dimension, whose Rad(A) is quotient finite dimensional, then these two dimensions for Rad(A) coincide. In particular, we prove that if an R-module A has finite hollow dimension, then A has small Krull dimension if and only if it has Krull dimension. Consequently, we show that if A has properties AB5* and qfd, then A has s.Krull dimension if and only if A has Krull dimension.

Some Characterizations of Modules via Essentially Small Submodules

Some Characterizations of Modules via Essentially Small Submodules

Generalization of CS condition

Let R be an associative ring with identity. An R-module M is called an NCS module if C (M)∩S(M) = {0}, where C (M) and S(M) denote the set of all closed submodules and the set of all small submodules of M, respectively. It is clear that the NCS condition is a generalization of the well-known CS condition. Properties of the NCS conditions of modules and rings are explored in this article. In the end, it is proved that a ring R is right Σ-CS if and only if R is right perfect and right countably Σ-NCS. Recall that a ring R is called right Σ-CS if every direct sum of copies of R R is a CS module. And a ring R is called right countably Σ-NCS if every direct sum of countable copies of R R is an NCS module.

Some Results on δ-Semiperfect Rings and δ-Supplemented Modules

In [9], the author extends the definition of lifting and supplemented modules to -lifting and -supplemented by replacing small submodule with submodule introduced by Zhou in [13]. The aim of this paper is to show new properties of -lifting and -supplemented modules. Especially, we show that any finite direct sum of -hollow modules is -supplemented. On the other hand, the notion of amply -supplemented modules is studied as a generalization of amply supplemented modules and several properties of these modules are given. We also prove that a module M is Artinian if and only if M is amply -supplemented and satisfies Descending Chain Condition (DCC) on -supplemented modules and on -small submodules. Finally, we obtain the following result: a ring R is right Artinian if and only if R is a -semiperfect ring which satisfies DCC on -small right ideals of R.

A note on localization of supplemented modules

In this paper we study on commutative rings with identity and all modules are unital left R-modules unless otherwise stated. We define the concept of small submodules for localization modules and additionally, we present the relation between an R-module M and an R_p-localization module M_p for all maximal ideals of R in view of being supplemented.

The Proper Class Generated by Weak Supplements

We show that, for hereditary rings, the smallest proper classes containing respectively the classes of short exact sequences determined by small submodules, submodules that have supplements and weak supplement submodules coincide. Moreover, we show that this class can be obtained as a natural extension of the class determined by small submodules. We also study injective, projective, coinjective and coprojective objects of this class. We prove that it is coinjectively generated and its global dimension is at most 1. Finally, we describe this class for Dedekind domains in terms of supplement submodules.