Simple-direct-injective modules

Camillo, Victor; Ibrahim, Yasser; Yousif, Mohamed; Zhou, Yiqiang

doi:10.1016/j.jalgebra.2014.07.033

Cited by 20 publications

(23 citation statements)

References 2 publications

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“…Thus R is right Noetherian. Now, by the same proof of (1) ⇒ (3) of Theorem 3.4 in [5], one infers that R is an Artinian serial ring with J(R) 2 = 0.…”

Section: Semisimple-direct-injective Modulesmentioning

confidence: 72%

“…However, C3 is a weaker property in general: if R is any integral domain which is not a field, then R is C3, but not C2. Recently, the classes of Ci-modules (i = 1, 2, 3) are studied and generalizations of them are considered ( [1,5,6,11,13,15]).…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Camillo, Ibrahim, Yousif and Zhou [5] obtained that every simple submodule which is isomorphic to a direct summand is itself a direct summand if and only if the sum of any two simple direct summands with zero intersection is again a direct summand [5,Proposition 2.1]. Such modules are called simple-direct-injective (see also [11]).…”

Section: Introductionmentioning

confidence: 99%

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Semisimple-direct-injective modules

Abyzov

Koşan

Quynh

et al. 2021

Hacettepe Journal of Mathematics and Statistics

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The notion of simple-direct-injective modules which are a generalization of injective modules unifies C2 and C3-modules. In the present paper, we introduce the notion of the semisimple-direct-injective module which gives a unified viewpoint of C2, C3, SSP properties and simple-direct-injective modules. It is proved that a ring R is Artinian serial with the Jacobson radical square zero if and only if every semisimple-direct-injective right Rmodule has the SSP and, for any family of simple injective right R-modules {S i } I , ⊕ I S i is injective. We also show that R is a right Noetherian right V-ring if and only if every right R-module has a semisimple-direct-injective envelope if and only if every right R-module has a semisimple-direct-injective cover.

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“…Thus R is right Noetherian. Now, by the same proof of (1) ⇒ (3) of Theorem 3.4 in [5], one infers that R is an Artinian serial ring with J(R) 2 = 0.…”

Section: Semisimple-direct-injective Modulesmentioning

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Semisimple-direct-injective modules

Abyzov

Koşan

Quynh

et al. 2021

Hacettepe Journal of Mathematics and Statistics

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“…The following are equivalent for a ring R.(1) R is a right noetherian right V-ring;(Every simple continuous module is strongly Soc-injective.Proof. Similar to[5, Proposition 4.3].…”

mentioning

confidence: 76%

“…The authors [5] proved that a ring R is a right V -ring if and only if every right R-module is a simple-direct-injective module. As a generalization of V -rings, the notion of GV -rings was posed by V. S. Ramamurthi, K. M. Rangaswamy [14].…”

Section: Singular Simple-direct-injective Modules and Gv-ringsmentioning

confidence: 99%

Simple continuous modules

Wang¹

2018

HJMS

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A module M is called a simple continuous module if it satisfies the conditions (min − C1) and (min − C2). A module M is called singular simple-direct-injective if for any singular simple submodules A, B of M with A ∼ = B | M , then A | M. Various basic properties of these modules are proved, and some well-studied rings are characterized using simple continuous modules and singular simple-direct-injective modules. For instance, it is shown that a ring R is a right V-ring if and only if every right R-module is a simple continuous modules, and that a regular ring R is a right GV-ring if and only if every cyclic right R-module is a singular simple-direct-injective module.

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