weakly supplement extending modules

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.

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Strongly Weakly Supplement Extending Modules

Al-Mustansiriyah Journal of Science ◽

10.23851/mjs.v30i4.682 ◽

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.

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

Glasnik Matematicki ◽

10.3336/gm.56.2.08 ◽

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.

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Characterization of Self-Adjoint Domains for Two-Interval Even Order Singular C-Symmetric Differential Operators in Direct Sum Spaces

Discrete Dynamics in Nature and Society ◽

10.1155/2019/3210983 ◽

2019 ◽

Vol 2019 ◽

pp. 1-12

Author(s):

Qinglan Bao ◽

Xiaoling Hao ◽

Jiong Sun

Keyword(s):

Boundary Conditions ◽

Differential Operators ◽

Direct Sum ◽

Even Order ◽

Special Case ◽

Symmetric Differential Operators

This paper is concerned with the characterization of all self-adjoint domains associated with two-interval even order singular C-symmetric differential operators in terms of boundary conditions. The previously known characterizations of Lagrange symmetric differential operators are a special case of this one.

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The 2-primitive ideals of structural matrix near-rings

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500007136 ◽

1991 ◽

Vol 34 (2) ◽

pp. 229-239 ◽

Cited By ~ 2

Author(s):

L. Van Wyk

Keyword(s):

Direct Sum ◽

Primitive Ideals ◽

Structural Matrix

An isomorphism (as groups) is established between an arbitrary connected module over a structural matrix near-ring and a direct sum of appropriate modules over the base near-ring. This isomorphism leads to a characterization of the 2-primitive ideals of a structural matrix near-ring.

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RATIONALITY PROPERTIES OF MANIFOLDS CONTAINING QUASI-LINES

International Journal of Mathematics ◽

10.1142/s0129167x03002095 ◽

2003 ◽

Vol 14 (10) ◽

pp. 1053-1080 ◽

Cited By ~ 10

Author(s):

PALTIN IONESCU ◽

DANIEL NAIE

Keyword(s):

Open Subset ◽

Basic Question ◽

Projective Manifold ◽

Rational Varieties ◽

Direct Sum ◽

Formal Geometry ◽

Rationally Connected ◽

Birational Invariant ◽

Small Deformations

Let X be a complex, rationally connected, projective manifold. We show that X admits a modification [Formula: see text] that contains a quasi-line, i.e. a smooth rational curve whose normal bundle is a direct sum of copies of [Formula: see text]. For manifolds containing quasi-lines, a sufficient condition of rationality is exploited: there is a unique quasi-line from a given family passing through two general points. We define a numerical birational invariant, e(X), and prove that X is rational if and only if e(X)=1. If X is rational, there is a modification [Formula: see text] which is strongly-rational, i.e. contains an open subset isomorphic to an open subset of the projective space whose complement is at least 2-codimensional. We prove that strongly-rational varieties are stable under smooth, small deformations. The argument is based on a convenient characterization of these varieties. Finally, we relate the previous results and formal geometry. This relies on ẽ(X, Y), a numerical invariant of a given quasi-line Y that depends only on the formal completion [Formula: see text]. As applications we show various instances in which X is determined by [Formula: see text]. We also formulate a basic question about the birational invariance of ẽ(X, Y).

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A characterization of noetherian rings by cyclic modules

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500022987 ◽

1996 ◽

Vol 39 (2) ◽

pp. 253-262 ◽

Cited By ~ 3

Author(s):

Dinh Van Huynh

Keyword(s):

Projective Module ◽

Noetherian Rings ◽

Noetherian Module ◽

Direct Sum

It is shown that a ring R is right noetherian if and only if every cyclic right R-module is injective or a direct sum of a projective module and a noetherian module.

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On the Spezialschar of Maass

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2010/726549 ◽

2010 ◽

Vol 2010 ◽

pp. 1-15 ◽

Cited By ~ 8

Author(s):

Bernhard Heim

Keyword(s):

Modular Forms ◽

Siegel Modular Forms ◽

Special Values ◽

Direct Sum

Let be the space of Siegel modular forms of degree and even weight . In this paper firstly a certain subspace Spez the Spezialschar of , is introduced. In the setting of the Siegel threefold, it is proven that this Spezialschar is the Maass Spezialschar. Secondly, an embedding of into a direct sum Sym2is given. This leads to a basic characterization of the Spezialschar property. The results of this paper are directly related to the nonvanishing of certain special values of L-functions related to the Gross-Prasad conjecture. This is illustrated by a significant example in the paper.

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On coseparable and 𝔪-coseparable modules

Journal of Algebra and Its Applications ◽

10.1142/s0219498821500316 ◽

2020 ◽

pp. 2150031

Author(s):

Rachid Ech-chaouy ◽

Abdelouahab Idelhadj ◽

Rachid Tribak

Keyword(s):

Direct Summand ◽

Noetherian Rings ◽

Direct Products ◽

Finitely Generated ◽

Direct Sums ◽

Direct Sum ◽

Free Modules

A module [Formula: see text] is called coseparable ([Formula: see text]-coseparable) if for every submodule [Formula: see text] of [Formula: see text] such that [Formula: see text] is finitely generated ([Formula: see text] is simple), there exists a direct summand [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is finitely generated. In this paper, we show that free modules are coseparable. We also investigate whether or not the ([Formula: see text]-)coseparability is stable under taking submodules, factor modules, direct summands, direct sums and direct products. We show that a finite direct sum of coseparable modules is not, in general, coseparable. But the class of [Formula: see text]-coseparable modules is closed under finite direct sums. Moreover, it is shown that the class of coseparable modules over noetherian rings is closed under finite direct sums. A characterization of coseparable modules over noetherian rings is provided. It is also shown that every lifting (H-supplemented) module is coseparable ([Formula: see text]-coseparable).

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Modules whose closed submodules are finitely generated

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500005083 ◽

1991 ◽

Vol 34 (1) ◽

pp. 161-166 ◽

Cited By ~ 3

Author(s):

Nguyen V. Dung

Keyword(s):

CS-modules and annihilator conditions

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171203206219 ◽

2003 ◽

Vol 2003 (50) ◽

pp. 3195-3202 ◽

Cited By ~ 1

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.

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