scholarly journals Semi-T-Hollow Modules and Semi-T-Lifting Modules

2021 ◽  
pp. 2357-2361
Author(s):  
Alaa A. Elewi
Keyword(s):  
Direct Summand ◽  
Associative Ring ◽  
Left Module ◽  
Small Submodule

Let be an associative ring with identity and let be a unitary left -module. Let  be a non-zero submodule of .We say that  is a semi- - hollow module if for every submodule  of  such that  is a semi- - small submodule ( ). In addition, we say that  is a semi- - lifting module if for every submodule  of , there exists a direct summand  of  and  such that   The main purpose of this work was to develop the properties of these classes of module.  

scholarly journals ON M- Hollow modules

2010 ◽  
Vol 7 (4) ◽  
pp. 1442-1446
Author(s):  
Baghdad Science Journal
Keyword(s):  
Associative Ring ◽  
Left Module ◽  
Maximal Submodule ◽  
Small Submodule

Let R be associative ring with identity and M is a non- zero unitary left module over R. M is called M- hollow if every maximal submodule of M is small submodule of M. In this paper we study the properties of this kind of modules.

Zariski-like topologies for lattices with applications to modules over associative rings

2019 ◽  
Vol 18 (07) ◽  
pp. 1950131
Author(s):  
Jawad Abuhlail ◽  
Hamza Hroub
Keyword(s):  
Complete Lattice ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Proper Class ◽  
Topological Properties ◽  
Left Module ◽  
Separation Axioms ◽  
Prime Submodules ◽  
Algebraic Properties ◽  
Associative Rings

We study Zariski-like topologies on a proper class [Formula: see text] of a complete lattice [Formula: see text]. We consider [Formula: see text] with the so-called classical Zariski topology [Formula: see text] and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for it to be spectral. We say that [Formula: see text] is [Formula: see text]-top if [Formula: see text] is a topology. We study the interplay between the algebraic properties of an [Formula: see text]-top complete lattice [Formula: see text] and the topological properties of [Formula: see text] Our results are applied to several spectra which are proper classes of [Formula: see text] where [Formula: see text] is a nonzero left module over an arbitrary associative ring [Formula: see text] (e.g. the spectra of prime, coprime, fully prime submodules) of [Formula: see text] as well as to several spectra of the dual complete lattice [Formula: see text] (e.g. the spectra of first, second and fully coprime submodules of [Formula: see text]).

scholarly journals On p-injective rings

1995 ◽  
Vol 37 (3) ◽  
pp. 373-378 ◽  
Author(s):  
Gennadi Puninski ◽  
Robert Wisbauer ◽  
Mohamed Yousif
Keyword(s):  
Direct Summand ◽  
Jacobson Radical ◽  
Associative Ring ◽  
The Right ◽  
Left Annihilator

Throughout this paper R will be an associative ring with unity and all R-modules are unitary. The right (resp. left) annihilator in R of a subset X of a module is denoted by r(X)(resp. I(X)). The Jacobson radical of R is denoted by J(R), the singular ideals are denoted by Z(RR) and Z(RR) and the socles by Soc(RR) and Soc(RR). For a module M, E(M) and PE(M) denote the injective and pure-injective envelopes of M, respectively. For a submodule A ⊆ M, the notation A ⊆⊕M will mean that A is a direct summand of M.

scholarly journals Doily as Subgeometry of a Set of Nonunimodular Free Cyclic Submodules

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 28
Author(s):  
Metod Saniga ◽  
Edyta Bartnicka
Keyword(s):  
Quantum Information ◽  
Geometric Structure ◽  
Associative Ring ◽  
Generalized Quadrangle ◽  
Two Dimensional ◽  
Left Module ◽  
Free Left ◽  
Cyclic Submodules ◽  
Free Cyclic Submodules

In this paper, it is shown that there exists a particular associative ring with unity of order 16 such that the relations between non-unimodular free cyclic submodules of its two-dimensional free left module can be expressed in terms of the structure of the generalized quadrangle of order two. Such a doily-centered geometric structure is surmised to be of relevance for quantum information.

scholarly journals On Quasi-h-Dense Submodules and h-Pure Envelopes of QTAG Modules

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Alveera Mehdi ◽  
Fahad Sikander ◽  
Firdhousi Begum
Keyword(s):  
Direct Summand ◽  
Homomorphic Image ◽  
Associative Ring ◽  
Finitely Generated ◽  
Direct Sum ◽  
Uniserial Modules ◽  
Pure Submodules

A module M over an associative ring R with unity is a QTAG module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. There are many fascinating properties of QTAG modules of which h-pure submodules and high submodules are significant. A submodule N is quasi-h-dense in M if M/K is h-divisible, for every h-pure submodule K of M, containing N. Here we study these submodules and obtain some interesting results. Motivated by h-neat envelope, we also define h-pure envelope of a submodule N as the h-pure submodule K⊇N if K has no direct summand containing N. We find that h-pure envelopes of N have isomorphic basic submodules, and if M is the direct sum of uniserial modules, then all h-pure envelopes of N are isomorphic.

scholarly journals On Jacobson – Small Submodules

2019 ◽  
pp. 1584-1591
Author(s):  
Ali Kabban ◽  
Wasan Khalid
Keyword(s):  
Associative Ring ◽  
Direct Sum ◽  
Small Submodule

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.

Direct Sums of Hollow-lifting Modules

2012 ◽  
Vol 19 (01) ◽  
pp. 87-97
Author(s):  
Yongduo Wang ◽  
Dejun Wu
Keyword(s):  
Direct Summand ◽  
Direct Sums ◽  
Direct Sum ◽  
Small Submodule

A module M over a ring R is called hollow-lifting if every submodule N of M with M/N hollow contains a direct summand K of M such that N/K is a small submodule of M/K. It is known that a (finite) direct sum of hollow-lifting modules need not be hollow-lifting. In this paper, we show that a quotient of a hollow-lifting module is not hollow-lifting in general, and discuss when a direct sum of hollow-lifting modules is hollow-lifting. We also study relative properties of hollow-lifting modules and investigate direct sums of X-hollow-lifting modules.

scholarly journals Strongly K-nonsingular Modules

2019 ◽  
Vol 32 (1) ◽  
pp. 173
Author(s):  
Tha'ar Younis Ghawi
Keyword(s):  
Direct Summand ◽  
Direct Sums ◽  
Nonzero Intersection ◽  
Small Submodule

       A submodule N of a module M  is said to be s-essential if it has nonzero intersection with any nonzero small submodule in M. In this article, we introduce and study a class of modules in which all its nonzero endomorphisms have non-s-essential kernels, named, strongly -nonsigular. We investigate some properties of strongly -nonsigular modules. Direct summand, direct sums and some connections of such modules are discussed.        

Cline’s formula for g-Drazin inverses in a ring

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2249-2255
Author(s):  
Huanyin Chen ◽  
Marjan Abdolyousefi
Keyword(s):  
Operator Theory ◽  
Spectral Properties ◽  
Associative Ring ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Cline’S Formula

It is well known that for an associative ring R, if ab has g-Drazin inverse then ba has g-Drazin inverse. In this case, (ba)d = b((ab)d)2a. This formula is so-called Cline?s formula for g-Drazin inverse, which plays an elementary role in matrix and operator theory. In this paper, we generalize Cline?s formula to the wider case. In particular, as applications, we obtain new common spectral properties of bounded linear operators.

Extensions of Rings Having McCoy Condition

2009 ◽  
Vol 52 (2) ◽  
pp. 267-272 ◽  
Author(s):  
Muhammet Tamer Koşan
Keyword(s):  
Positive Integer ◽  
Associative Ring ◽  
Nonzero Element ◽  
Ring Extensions

AbstractLet R be an associative ring with unity. Then R is said to be a right McCoy ring when the equation f (x)g(x) = 0 (over R[x]), where 0 ≠ f (x), g(x) ∈ R[x], implies that there exists a nonzero element c ∈ R such that f (x)c = 0. In this paper, we characterize some basic ring extensions of right McCoy rings and we prove that if R is a right McCoy ring, then R[x]/(xn) is a right McCoy ring for any positive integer n ≥ 2.

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