Extending modules

Dũng, Nguyễn Việt; Huynh, Dinh Van; Smith, Patrick F.; Wisbauer, Robert

doi:10.1201/9780203756331

Cited by 50 publications

(78 citation statements)

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“…Also, we have here a direct sum of weakly supplement extending modules need not necessary weakly supplement extending. M= Z[X] ⨁Z[X] as Z[X]-module is not weakly supplement extending, because M is not extending [3] and Rad(M)=0 [11]. Recall that, a module M is distributive if for all submodule A, S and N of M, A ∩ (S + N) = (A ∩ S)+ (A ∩ N) [12].…”

Section: Corollary (217)mentioning

confidence: 99%

“…If every submodule J of M is weakly supplement of M, then a module M is said to be weakly supplemented [2]. A module M is uniform if each nonzero submodule of M is essential submodule in M [3]. A module M is ⨁-supplemented, if for any submodule N of M has a supplement submodule that is direct summand [2].…”

Section: Introductionmentioning

confidence: 99%

“…The singular submodule of a module M is Z(M)={a ∈M | Ea=0 for some essential left ideal E of R}. A module M is said to be singular if Z(M)=M, and M is said to be non-singular if Z(M)=0 [3]. A module M is said to be lifting, if each submodule L of M there is a direct summand W of M with W⊆ L such that M=W⊕W' and ′ ∩ ≪W' [2].…”

Section: Introductionmentioning

confidence: 99%

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weakly supplement extending modules

Al-Saadi

musa

2020

MJS

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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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Section: Corollary (217)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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weakly supplement extending modules

Al-Saadi

musa

2020

MJS

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“…Modüller birimseldir ve bir değişmeli grup için sağ -modülü gösterecektir. Çalışmamızda açıklanmamış terminoloji ve aşağıda verilen tanımlara ilişkin ayrıntılı bilgi için [3] ve [5] kitapları okuyucuya önerilmektedir. Bir alt kümesinin (sırasıyla, bir x elemanının) bir -modülünde sağ sıfırlayıcısı (annihilator) ( ) (sırasıyla, ( )) ile gösterilecektir.…”

Section: Introductionunclassified

Characterization of Some Well-Known Rings by Using Sa Modules

Taşdemir¹

2019

Anadolu Üniversitesi Bilim Ve Teknoloji Dergisi - B Teorik Bilimler

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ÖZETBir R halkası üzerinde tanımlı bir M modülü, hem SIP hem de ADS özelliklerine sahipse, bu M modülüne SA modül dendiğini biliyoruz. Bu çalışmada, her SA modülün aynı zamanda SSP özelliğine de sahip olduğu kanıtlanmıştır. Ayrıca, eğer birmodül, injektif ve asal ise SSP özelliğine sahip olduğu gösterilmiştir. Bunlara ek olarak, çok bilinen bazı halkalar, SA modüller yardımıyla karakterize edilmiştir. ABSTRACTRecall that an R-module M is an SA module if M satisfies both SIP and ADS property. In this study, it is proved that every SA module has the SSP, and also proved that if an R-module M is both injective and prime then M has the SSP. Additionally, some well-known rings are characterized by using SA modules.

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“…the submodule of M , the essential submodule of M , the complement submodule of M , the direct summand of M and the projection invariant submodule of M , respectively.For unknown terminology and notation, see[1,4,10,11].…”

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confidence: 99%

Modules whose projection invariant submodules have projection invariant closures

Kara¹

2018

MMN

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We elaborate the class of P C-modules in which every projection invariant submodules has projection invariant closures in this study. We provide examples that the class of P Cmodules does not belong to the system of generalization of extending modules. Moreover, we clarify direct sums and direct summands properties for the former class of modules. It is proved that the aforementioned property is not closed under direct sums. Thereupon, we cope with when the direct sums of P C-modules enjoy with the property.

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Extending modules

Cited by 50 publications

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weakly supplement extending modules

weakly supplement extending modules

Characterization of Some Well-Known Rings by Using Sa Modules

Modules whose projection invariant submodules have projection invariant closures

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