Cyclic Submodules Research Articles

$S$-$M$-cyclic submodules and some applications

In this paper, we introduce the notion of $S$-$M$-cyclic submodules, which is a generalization of the notion of $M$-cyclic submodules. Let $M, N$ be right $R$-modules and $S$ be a multiplicatively closed subset of a ring $R$. A submodule $A$ of $N$ is said to be an $S$-$M$-cyclic submodule, if there exist $s\in S$ and $f \in Hom_R(M,N)$ such that $As \subseteq f(M) \subseteq A$. Besides giving many properties of $S$-$M$-cyclic submodules, we generalize some results on $M$-cyclic submodules to $S$-$M$-cyclic submodules. Furthermore, we generalize some properties of principally injective modules and pseudo-principally injective modules to $S$-principally injective modules and $S$-pseudo-principally injective modules, respectively. We study the transfer of this notion to various contexts of these modules.

A Note on Free Module Decomposition over A Principal Ideal Domain

Some methods have been used to express a finitely generated module over a principal ideal domain as a finite direct sum of its cyclic submodules. In this paper, we give an alternative technique to decompose a free module with finite rank over a principal ideal domain using eigen spaces of its endomorphism ring.

A Principally Radg-Lifting Modules

In this article we present a new class of modules which is named as a principally -lifting modules. This class termed by Principally -lifting in this work which defined as, a module is called Principally -lifting if for every cyclic submodule of with , there is a decomposition such that and is g-small in . Thus, a ring is called Principally -lifting if it is a principally -lifting as -module. We determined it is structure. Several characterizations, properties, and instances are described of these modules'.

Principally ⨁-G-〖Rad〗_g-supplemented modules

In this paper, a new concept has been presented that is a stronger than a previous concept called -G--supplemented. The basic definition is in an ring where an -module is said to be principally -G--supplemented (shortly, -PG--supplemented) if any cyclic submodule of with , there exists a direct summand of such that and . A set of properties and relations between previous modules and the given module has been dealt with simple examples illustrating those relations.

ON CYCLIC DECOMPOSITION OF Z-MODULE M_{m×r}(Z_n)

A torsion module over a principal ideal domain has special properties related to the way how it is decomposed either into primary or cyclic submodules. This paper carries out a special case of such module over the ring of integer, which consists of all matrices with entries from the set of integer modulo n. The result shows that its decomposition depends on the prime factors of n.

On Bass Modules and Semi-V-Modules

Let R be a commutative ring with identity. An $R$-module $M$ is called a semi-V-module if every nonzero homomorphic image of $M$ contains a nonzero V-submodule. An $R$-module $M$ is called a Bass module if every nonzero module in $\sigma [M]$ has a maximal submodule. It is shown that an $R$-module $M$ is a semi-V-module if and only if every nonzero cyclic submodule of $M$ is a Bass module if and only if $R/Ann_R(x)$ is a Bass ring for every nonzero element $x \in M$. Among other results, we characterize the class of rings $R$ for which every semi-V-module is a V-module and the class of rings $R$ for which every semi-V-module is a semisimple module. Also, we characterize the rings $R$ for which the class of semi-V-modules (Bass modules) is closed under direct products.

A note on almost prime submodule of CSM module over principal ideal domain

An almost prime submodule is a generalization of prime submodule introduced in 2011 by Khashan. This algebraic structure was brought from an algebraic structure in ring theory, prime ideal, and almost prime ideal. This paper aims to construct similar properties of prime ideal and almost prime ideal from ring theory to module theory. The problem that we want to eliminate is the multiplication operation, which is missing in module theory. We use the definition of module annihilator to bridge the gap. This article gives some properties of the prime submodule and almost prime submodule of CMS module over a principal ideal domain. A CSM module is a module that every cyclic submodule. One of the results is that the idempotent submodule is an almost prime submodule.

On a class of g-lifting modules

The aim of this note is to introduce and investigate a category of modules which is analogous to that of g-lifting and principally lifting modules. The module M is called principally g-lifting if for every cyclic submodule K of M there exists a decomposition M = A ⊕ B such that A is a submodule of K and K ∩ B is g-small in M. As an application, we defined principally g-semiperfect modules and characterize these modules in terms of principally g-lifting modules.

On atomic ideals in some factor rings of $C(X,\Bbb Z)$

A nonzero $R$-module $M$ is atomic if for each two nonzero elements $a, b$ in $M$, both cyclic submodules $Ra$ and $Rb$ have nonzero isomorphic submodules. In this article it is shown that for an infinite $P$-space $X$, the factor rings $C(X,\Bbb{Z})/C_F(X,\Bbb{Z})$ and $C_c(X)/C_F(X)$ have no atomic ideals. This fact generalizes a result published in paper by A. Mozaffarikhah, E. Momtahan, A. R. Olfati and S. Safaeeyan (2020), which says that for an infinite set $X$, the factor ring $\Bbb{Z}^X/ \Bbb{Z}^{(X)}$ has no atomic ideal. Another result is that for each infinite $P$-space $X$, the socle of the factor ring $C_c(X)/C_F(X)$ is always equal to zero. Also, zero-dimensional spaces $X$ are characterized for which $C^F(X,\Bbb{Z})/C_F(X,\Bbb{Z})$ have atomic ideals.

Orbits of Free Cyclic Submodules Over Rings of Lower Triangular Matrices

Given a ring T_n (ngeqslant 2) of lower triangular ntimes n matrices with entries from an arbitrary field F, we completely classify the orbits of free cyclic submodules of ^2T_n under the action of the general linear group GL_2(T_n). Interestingly, the total number of such orbits is found to be equal to the Bell number B_n. A representative of each orbit is also given.

J-semi regular modules

Let R be a ring with identity and let M be a left R-module. M is called J-semiregular module if every cyclic submodule of M is J-lying over a projective summand of M, The aim of this paper is to introduce properties of J-semiregular module Especially, we give characterizations of J-semiregular module. On the other hand, the notion of J-semi hollow modules is studied as a generalization of semi hollow modules, finally F-J-semiregular modules is studied as a generalization of F-semiregular modules.

Quotient-transitivity and cyclic submodule-transitivity for p-adic modules

Two new notions of transitivity, which we have named quotient-transitivity and transitivity with respect to cyclic submodules for $p$-adic modules, are introduced. Unlike the classical notions that derive from Abelian group theory, this approach is based

Modules of the 0-Hecke algebra arising from standard permuted composition tableaux

We study the Hn(0)-module Sασ due to Tewari and van Willigenburg, which was constructed using new combinatorial objects called standard permuted composition tableaux and decomposed into cyclic submodules. First, we show that every direct summand appearing in their decomposition is indecomposable and characterize when Sασ is indecomposable. Second, we find characteristic relations among Sασ's and expand the image of Sασ under the quasi characteristic in terms of quasisymmetric Schur functions. Finally, we show that the canonical submodule of Sασ appears as a homomorphic image of a projective indecomposable module.

D3-MODULES VERSUSD4-MODULES – APPLICATIONS TO QUIVERS

AbstractA moduleMis called aD4-module if, wheneverAandBare submodules ofMwithM=A⊕Bandf:A→Bis a homomorphism with Imfa direct summand ofB, then Kerfis a direct summand ofA. The class ofD4-modules contains the class ofD3-modules, and hence the class of semi-projective modules, and so the class of Rickart modules. In this paper we prove that, over a commutative Dedekind domainR, for anR-moduleMwhich is a direct sum of cyclic submodules,Mis direct projective (equivalently, it is semi-projective) iffMisD3 iffMisD4. Also we prove that, over a prime PI-ring, for a divisibleR-moduleX,Xis direct projective (equivalently, it is Rickart) iffX⊕XisD4. We determine someD3-modules andD4-modules over a discrete valuation ring, as well. We give some relevant examples. We also provide several examples onD3-modules andD4-modules via quivers.

On Modules Satisfying the Descending Chain Condition on Cyclic Submodules

A module satisfying the descending chain condition on cyclic submodules is called coperfect. The class of coperfect modules lies properly between the class of locally artinian modules and the class of semiartinian modules. Let R be a commutative ring with identity. We show that every semiartinian R-module is coperfect if and only if R is a T-ring. It is also shown that the class of coperfect R-modules coincides with the class of locally artinian R-modules if and only if 𝔪/𝔪2 is a finitely generated R-module for every maximal ideal 𝔪 of R.

On Singly Flat and Singly Injective Modules

Recall that a left module A (resp. right module B) is said to be singly injective (resp. singly flat) if $$Ext^{1}_{R}(F/K, A) = 0$$ (resp. $${\text {Tor}}_{1}^{R}(B, F/K)= 0$$ ) for any cyclic submodule K of any finitely generated free left R-module F. In this paper, we continue to study and investigate the homological objects related to singly flat and singly injective modules and module homomorphisms. Along the way, the right orthogonal class of singly flat right modules and the left orthogonal class of singly injective left modules are introduced and studied. These concepts are used to extend the some known results and to characterize pseudo-coherent rings and left singly injective rings. In terms of some derived functors, some homological dimensions are investigated. As applications, some new characterizations of von Neumann regular rings and left PP rings are given. Finally, we study the singly flatness and singly injectivity of homomorphism modules over a commutative ring.

On µ-Semiregular Module

Let R be an associative ring with identity and let M be right R-module M is called μ-semi hollow module if every finitely generated submodule of M is μ-small submodule of M The purpose of this paper is to give some properties of μ-semi hollow module. Also, we gives conditions under, which the direct sum of μ-semi hollow modules is μ-semi hollow. An R-module is said has a projective μ-cover if there exists an epimorphism f:P→M Where P is a projective R-module and ker (f)≪ P.And study some properties of Projective μ-cover of M. Were studied Moreover, An module M is μ-semiregular module if every cyclic submodule of M is μ-lying summand of M We add some results of μ-semiregular module.

On Divided Modules

Recall that a commutative ring R is said to be a divided ring if its each prime ideal P is comparable with each principal ideal (a), where $$a\in R.$$ In this paper, we extend the notion of divided rings to modules in two different ways: let R be a commutative ring with identity and M a unital R-module. Then M is said to be a divided (weakly divided) module if its each prime submodule N of M is comparable with each cyclic submodule Rm (rM) of M, where $$m\in M$$ ($$r\in R)$$. In addition to give many characterizations of divided modules, some topological properties of (quasi-) Zariski topology of divided modules are investigated. Also, we study the divided property of trivial extension $$R\ltimes M$$.

Affine and projective planes linked with projective lines over certain rings of lower triangular matrices

Let Tn(q) be the ring of lower triangular matrices of order n≥2 with entries from the finite field F(q) of order q≥2 and let Tn2(q) denote its free left module. For n=2,3 it is shown that the projective line over Tn(q) gives rise to a set of (q+1)(n−1)q3(n−1)(n−2)2 affine planes of order q. The points of such an affine plane are non-free cyclic submodules of Tn2(q) not contained in any non-unimodular free cyclic submodule of Tn2(q) and its lines are points of the projective line. Furthermore, it is demonstrated that each affine plane can be extended to the projective plane of order q, with the ‘line at infinity’ being represented by those free cyclic submodules of Tn2(q) that are generated by non-unimodular pairs. Our approach can be adjusted to address the case of arbitrary n.

Two Notes on Automorphisms of Modules over a PID

Let D be a principal ideal domain (PID) and M be a module over D. We prove the following two dual results: (i) If M is finitely generated and x, y are two elements in M such that [Formula: see text], then there exists an automorphism α of M such that [Formula: see text]. (ii) If M satisfies the minimal condition on submodules and X, Y are two locally cyclic submodules of M such that [Formula: see text] and [Formula: see text], then there exists an automorphism α of M such that [Formula: see text].