Strongly radical supplemented modules

Büyükaşık, Engı̇n; Türkmen, Ergül

doi:10.1007/s11253-012-0579-3

Cited by 10 publications

(9 citation statements)

References 5 publications

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“…Recall from [1] that a module M is called strongly radical supplemented if every submodule N of M containing Rad.M / has a supplement in M . It is proven in [1, Corollary 2.1] that finite sums of strongly radical supplemented modules are strongly radical supplemented.…”

Section: E -Modules and Ee -Modulesmentioning

confidence: 99%

Modules that have a supplement in every coatomic extension

Türkmen¹

2015

MMN

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Let R be a ring and M be an R-module. M is said to be an E-module (respectively, an EE-module) if M has a supplement (respectively, ample supplements) in every coatomic extension N , i.e. N M is coatomic. We prove that if a module M is an EE-module, every submodule of M is an E-module, and then we show that a ring R is left perfect iff every left R-module is an E-module iff every left R-module is an EE-module. We also prove that the class of E-modules is closed under extension. In addition, we give a new characterization of left V-rings by cofinitely injective modules.

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Section: E -Modules and Ee -Modulesmentioning

confidence: 99%

Modules that have a supplement in every coatomic extension

Türkmen¹

2015

MMN

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“…These modules are also a proper generalization of supplemented modules. Then, in [2], it is defined as a module M strongly radical supplemented (or briefly srs) if every submodule N of M with Rad(M ) ⊆ N has a supplement in M . Then it is introduced that modules whose every submodule containing the radical has a weak supplement (in particular, over dedekind domains the radical has a weak supplement) in the module as weakly radical supplemented module (wrs) which is a generalization of strongly radical supplemented modules [7].…”

Section: Introductionmentioning

confidence: 99%

Weakly Locally Artinian Supplemented Modules

Türkmen

2021

Journal of Universal Mathematics

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In this study, by using the concept of locally artinian supplemented modules, we have obtained weakly locally artinian supplemented modules as a proper generalization of these modules in module theory. Our results generalize and extend various comparable results in the existing literature. We have proved that the notion of weakly locally artinian supplemented modules inherited by factor modules, finite sums and small covers. We have obtained that weakly locally artinian supplemented modules with small radical coincide with weakly (radical) supplemented modules which have locally artinian radical. Also, we have shown that if N and M N are weakly locally artinian supplemented for some submodule N ⊆ M which has a weak locally artinian supplement in M then M is weakly locally artinian supplemented.

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“…He determined the structure of these modules over local Dedekind domains. Büyükaşık and Türkmen called a module M strongly radical supplemented (for shortly srs) if every submodule N of M with Rad(M ) ⊆ N have a supplement K in M (see [5]). They gave the various properties of srs-modules in the same paper.…”

Section: Introductionmentioning

confidence: 99%

“…They gave the various properties of srs-modules in the same paper. In particular, it was shown in [5,Proposition 2.3] that every finite sum of srs-modules is srs. By [5,Proposition 3.3], over a local Dedekind domain a module is radical supplemented if and only if it is srs.…”

Section: Introductionmentioning

confidence: 99%

Generalizations of Rad-supplemented modules

Kaynar¹,

Türkmen²,

Aydin³

2018

Publ Inst Math (Belgr)

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Let R be an associative ring with identity. We introduce the notion of semi-τ-supplemented modules, which is adapted from srs-modules, for a preradical τ on R-Mod. We provide basic properties of these modules. In particular, we study the objects of R-Mod for τ = Rad. We show that the class of semi-τ-supplemented modules is closed under finite sums and factor modules. We prove that, for an idempotent preradical τ on R-Mod, a module M is semi-τ-supplemented if and only if it is τ-supplemented. For τ = Rad, over a local ring every left module is semi-Rad-supplemented. We also prove that a commutative semilocal ring whose semi-Rad-supplemented modules are a direct sum of w-local left modules is an artinian principal ideal ring.

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Strongly radical supplemented modules

Cited by 10 publications

References 5 publications

Modules that have a supplement in every coatomic extension

Modules that have a supplement in every coatomic extension

Weakly Locally Artinian Supplemented Modules

Generalizations of Rad-supplemented modules

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