scholarly journals Strongly Weakly Supplement Extending Modules

2020 ◽  
Vol 30 (4) ◽  
pp. 71
Author(s):  
Aya Adnan Musa ◽  
Saad A. Al-Saadi
Keyword(s):  
Direct Sum ◽  
Closed Submodule ◽  
Extending Module ◽  
Supplement Submodule ◽  
Strong Concept

In this paper, a class of modules which are proper strong concept of weakly supplement extending modules will be introduced and studied. We call a module M is strongly weakly supplement extending, if each submodule of M is essential in fully invariant weakly supplement submodule in M. Many characterizations of strongly weakly supplement extending modules are obtained. We show that M is strongly weakly supplement extending module if and only if every closed submodule of M is fully invariant weakly supplement submodule in M. Also we study the relation among this concept and other known concepts of modules. Moreover, we give some conditions that of strongly weakly supplement extending modules is closed under direct sum property is strongly weakly supplement extending.

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. 

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.

Lifting modules with finite internal exchange property and direct sums of hollow modules

2020 ◽  
pp. 2150189
Author(s):  
Yosuke Kuratomi
Keyword(s):  
Direct Summand ◽  
Sufficient Conditions ◽  
Exchange Property ◽  
Direct Sums ◽  
Perfect Ring ◽  
Direct Sum ◽  
Decomposition Formula ◽  
Direct Sum Decomposition ◽  
Extending Module ◽  
Necessary And Sufficient

A module [Formula: see text] is said to be lifting if, for any submodule [Formula: see text] of [Formula: see text], there exists a decomposition [Formula: see text] such that [Formula: see text] and [Formula: see text] is a small submodule of [Formula: see text]. A lifting module is defined as a dual concept of the extending module. A module [Formula: see text] is said to have the finite internal exchange property if, for any direct summand [Formula: see text] of [Formula: see text] and any finite direct sum decomposition [Formula: see text], there exists a direct summand [Formula: see text] of [Formula: see text] [Formula: see text] such that [Formula: see text]. This paper is concerned with the following two fundamental unsolved problems of lifting modules: “Classify those rings all of whose lifting modules have the finite internal exchange property” and “When is a direct sum of indecomposable lifting modules lifting?”. In this paper, we prove that any [Formula: see text]-square-free lifting module over a right perfect ring satisfies the finite internal exchange property. In addition, we give some necessary and sufficient conditions for a direct sum of hollow modules over a right perfect ring to be lifting with the finite internal exchange property.

scholarly journals Nonsingular bilinear forms on direct sums of ideals

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Beata Rothkegel
Keyword(s):  
Bilinear Form ◽  
Dedekind Domain ◽  
Bilinear Forms ◽  
Direct Sums ◽  
Direct Sum

AbstractIn the paper we formulate a criterion for the nonsingularity of a bilinear form on a direct sum of finitely many invertible ideals of a domain. We classify these forms up to isometry and, in the case of a Dedekind domain, up to similarity.

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

scholarly journals Tropical ideals do not realise all Bergman fans

2021 ◽  
Vol 8 (3) ◽  
Author(s):  
Jan Draisma ◽  
Felipe Rincón
Keyword(s):  
Tropical Geometry ◽  
Tensor Products ◽  
Tropical Variety ◽  
Polyhedral Complex ◽  
Direct Sum ◽  
Weight One ◽  
Polyhedral Complexes

AbstractEvery tropical ideal in the sense of Maclagan–Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank two on three elements and in which all maximal cones have weight one.

scholarly journals Singular Bogoliubov transformations and inequivalent representations of canonical commutation relations

2019 ◽  
Vol 31 (08) ◽  
pp. 1950026 ◽  
Author(s):  
Asao Arai
Keyword(s):  
Fock Space ◽  
Sufficient Conditions ◽  
Bogoliubov Transformation ◽  
Fock Representation ◽  
Canonical Commutation Relations ◽  
Commutation Relations ◽  
Direct Sum ◽  
Boson Fock Space ◽  
Inequivalent Representations ◽  
Bogoliubov Transformations

We introduce a concept of singular Bogoliubov transformation on the abstract boson Fock space and construct a representation of canonical commutation relations (CCRs) which is inequivalent to any direct sum of the Fock representation. Sufficient conditions for the representation to be irreducible are formulated. Moreover, an example of such representations of CCRs is given.

scholarly journals Direct Sum Cancellation of Noetherian Modules

1998 ◽  
Vol 200 (1) ◽  
pp. 207-224 ◽  
Author(s):  
Gary Brookfield
Keyword(s):  
Direct Sum ◽  
Noetherian Modules

scholarly journals CHARACTER CLUSTERS FOR LIE ALGEBRA MODULES OVER A FIELD OF NONZERO CHARACTERISTIC

2013 ◽  
Vol 89 (2) ◽  
pp. 234-242 ◽  
Author(s):  
DONALD W. BARNES
Keyword(s):  
Lie Algebra ◽  
Saturated Formation ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Nonzero Characteristic ◽  
Composition Factors

AbstractFor a Lie algebra $L$ over an algebraically closed field $F$ of nonzero characteristic, every finite dimensional $L$-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra $L$ is in the saturated formation $\mathfrak{F}$ and if $V, W$ are irreducible $L$-modules with the same cluster and the $p$-operation vanishes on the centre of the $p$-envelope used, then $V, W$ are either both $\mathfrak{F}$-central or both $\mathfrak{F}$-eccentric. Clusters are used to generalise the construction of induced modules.

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