ON A MODULE CONTAINING A COPY OF ITS FACTOR BY A t-CLOSED SUBMODULE

2020 ◽  
pp. 111-126
Author(s):  
Mouhamadou Lamine Dia ◽  
Papa Cheikhou Diop ◽  
Abdoul Djibril Diallo ◽  
Mamadou Barry
Keyword(s):  
Closed Submodule

Hilbert \(C^{*}\)-modules in which all relatively strictly closed submodules are complemented

2021 ◽  
Vol 56 (2) ◽  
pp. 343-374
Author(s):  
Boris Guljaš ◽  
Keyword(s):  
Hilbert Space ◽  
Hilbert Modules ◽  
Strict Topology ◽  
Direct Sums ◽  
Essential Ideal ◽  
Closed Submodule ◽  
Closed Submodules ◽  
Hereditary Algebras ◽  
Multiplier Algebras

We give the characterization and description of all full Hilbert modules and associated algebras having the property that each relatively strictly closed submodule is orthogonally complemented. A strict topology is determined by an essential closed two-sided ideal in the associated algebra and a related ideal submodule. It is shown that these are some modules over hereditary algebras containing the essential ideal isomorphic to the algebra of (not necessarily all) compact operators on a Hilbert space. The characterization and description of that broader class of Hilbert modules and their associated algebras is given. As auxiliary results we give properties of strict and relatively strict submodule closures, the characterization of orthogonal closedness and orthogonal complementing property for single submodules, relation of relative strict topology and projections, properties of outer direct sums with respect to the ideals in \(\ell_\infty\) and isomorphisms of Hilbert modules, and we prove some properties of hereditary algebras and associated hereditary modules with respect to the multiplier algebras, multiplier Hilbert modules, corona algebras and corona modules.

scholarly journals Modules with Chain Conditions on S-Closed Submodules

2017 ◽  
Vol 30 (3) ◽  
pp. 227
Author(s):  
Rana Noori Majeed Mohammed
Keyword(s):  
Commutative Ring ◽  
Essential Extension ◽  
Chain Conditions ◽  
Closed Submodule ◽  
Closed Submodules

  Let L be a commutative ring with identity and let W be a unitary left L- module. A submodule D of an L- module W is called  s- closed submodule denoted by  D ≤sc W, if D has   no  proper s- essential extension in W, that is , whenever D ≤ W such that D ≤se H≤ W, then D = H. In  this  paper,  we study  modules which satisfies  the ascending chain  conditions (ACC) and descending chain conditions (DCC) on this kind of submodules.

When some complement of a z-closed submodule is a summand

2017 ◽  
Vol 46 (7) ◽  
pp. 3071-3078 ◽  
Author(s):  
Yeliz Kara ◽  
Adnan Tercan
Keyword(s):  
Closed Submodule

On modules in which every weak large closed submodule is a direct summand

2018 ◽  
Vol 12 (23) ◽  
pp. 1109-1119
Author(s):  
M. H. Elbaroudy ◽  
M. A. Kamal ◽  
Manar E. Tabarak
Keyword(s):  
Direct Summand ◽  
Closed Submodule

scholarly journals Modules whose closed submodules are finitely generated

1991 ◽  
Vol 34 (1) ◽  
pp. 161-166 ◽  
Author(s):  
Nguyen V. Dung
Keyword(s):  
Recent Result ◽  
Cyclic Module ◽  
Finitely Generated ◽  
Direct Sum ◽  
Cyclic Submodules ◽  
Closed Submodule ◽  
Closed Submodules

A module M is called a CC-module if every closed submodule of M is cyclic. It is shown that a cyclic module M is a direct sum of indecomposable submodules if all quotients of cyclic submodules of M are CC-modules. This theorem generalizes a recent result of B. L. Osofsky and P. F. Smith on cyclic completely CS-modules. Some further applications are given for cyclic modules which are decomposed into projectives and injectives.

scholarly journals CS-modules and annihilator conditions

2003 ◽  
Vol 2003 (50) ◽  
pp. 3195-3202 ◽  
Author(s):  
Mahmoud A. Kamal ◽  
Amany M. Menshawy
Keyword(s):  
Direct Connection ◽  
Closed Submodule

We studyS−R-bimodulesSMRwith the annihilator conditionS=lS(A)+lS(B)for any closed submoduleA, and a complementBofA, inMR. Such annihilator condition has a direct connection with the CS-condition forMR. We make use of this to give a new characterization of CS-modules. BimodulesSMRfor whichrMlS(A)=A(for every closed submoduleAofMR) are also dealt with. Such modules are calledW∗-modules. We give the extra added annihilator conditions toW∗-modules to be equivalent to the continuous (quasicontinuous) modules.

scholarly journals W-Closed Submodule and Related Concepts

2018 ◽  
Vol 31 (2) ◽  
pp. 164
Author(s):  
Haibat K. Mohammad Ali ◽  
Mohammad E. Dahsh
Keyword(s):  
Commutative Ring ◽  
Chain Condition ◽  
Essential Extension ◽  
Basic Properties ◽  
Closed Submodule ◽  
Closed Submodules

    Let R be a commutative ring with identity, and M be a left untial module. In this paper we introduce and study the concept w-closed submodules, that is stronger form of the concept of closed submodules, where asubmodule K of a module M is called w-closed in M, "if it has no proper weak essential extension in M", that is if there exists a submodule L of M with K is weak essential submodule of L then K=L. Some basic properties, examples of w-closed submodules are investigated, and some relationships between w-closed submodules and other related modules are studied. Furthermore, modules with chain condition on w-closed submodules are studied.   

Strongly C_11-Condition Modules and Strongly T_11-Type Modules

2018 ◽  
pp. 358
Author(s):  
Inaam M A Hadi ◽  
Farhan D Shyaa
Keyword(s):  
Direct Summand ◽  
Closed Submodule

      In this paper, we introduced module that satisfying strongly -condition modules and strongly -type modules as generalizations of t-extending. A module  is said strongly -condition if for every submodule of  has a complement which is fully invariant direct summand. A module   is said to be strongly -type modules if every t-closed submodule has a complement which is a fully invariant direct summand. Many characterizations for modules with strongly -condition for strongly -type module are given. Also many connections between these types of module and some related types of modules are investigated.

scholarly journals weakly supplement extending modules

2020 ◽  
Vol 31 (2) ◽  
pp. 38
Author(s):  
Saad A. Al-Saadi ◽  
Aya Adnan Musa
Keyword(s):  
Direct Sum ◽  
Closed Submodule ◽  
Extending Module ◽  
Supplement Submodule

In this paper, the extending property of modules is generalized by using weakly supplement submodules. We call a module M is weakly supplement extending if each submodule of M is essential in a weakly supplement submodule of M. Many characterization of weakly supplement extending module are obtained, we show that M is weakly supplement extending if and only if each closed submodule is weakly supplementing submodule of M. Moreover, we study the relation of weakly supplement extending module and among other known classes of the module such as lifting module, weakly supplemented module, supplement extending module and others. Also, we study conditions under it a direct sum of weakly supplement extending module is weakly supplement extending. 

ON M-C-INJECTIVE AND SELF-C-INJECTIVE MODULES

2010 ◽  
Vol 03 (03) ◽  
pp. 387-393 ◽  
Author(s):  
A. K. Chaturvedi ◽  
B. M. Pandeya ◽  
A. M. Tripathi ◽  
O. P. Mishra
Keyword(s):  
Endomorphism Ring ◽  
Projective Module ◽  
Injective Module ◽  
Injective Modules ◽  
Factor Module ◽  
Closed Submodule

Let M1 and M2 be two R-modules. Then M2 is called M1-c-injective if every homomorphism α from K to M2, where K is a closed submodule of M1, can be extended to a homomorphism β from M1 to M2. An R-module M is called self-c-injective if M is M-c-injective. For a projective module M, it has been proved that the factor module of an M -c-injective module is M -c-injective if and only if every closed submodule of M is projective. A characterization of self-c-injective modules in terms endomorphism ring of an R-module satisfying the CM-property is given.

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