On coseparable and 𝔪-coseparable modules

Ech-chaouy, Rachid; Idelhadj, Abdelouahab; Tribak, Rachid

doi:10.1142/s0219498821500316

Cited by 2 publications

(6 citation statements)

References 13 publications

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“…Case 2: Assume that M is not finitely generated. Note that M is coseparable by [8,Proposition 3.16]. Hence, M is s-coseparable by Theorem 2.7.…”

Section: Proposition 227 Letmentioning

confidence: 92%

“…Let N be a cofinite submodule of M. Then clearly N is not finitely generated. Note that N is coseparable by [8,Proposition 2.15]. Therefore, N is s-coseparable by Theorem 2.7.…”

Section: Corollary 217 Let R Be a Right Noetherian Ring And Let M Be An S-coseparable R-module Then Every Cofinite Submodule Of M Is Alsomentioning

confidence: 94%

“…Proof Note that M is coseparable. Then N is coseparable by [8,Proposition 2.12]. Since N is not finitely generated, it follows that N is s-coseparable by Theorem 2.7.…”

Section: Proposition 213mentioning

confidence: 97%

“…The following characterization provides a rich source of examples of s-coseparable modules (see [8]).…”

Section: Example 26mentioning

confidence: 99%

“…This concept was initially appeared in 1968 [9] for abelian groups. Then the concept was generalized to modules in 1978 [12] by Hiremath and was further studied by other authors (see e.g., [8,19]). In this paper, we focus on a subclass of coseparable modules.…”

Section: Introductionmentioning

confidence: 99%

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On strongly coseparable modules

Ech-chaouy

Idelhadj

Tribak³

2021

Arab. J. Math.

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A module M is called $$\mathfrak {s}$$ s -coseparable if for every nonzero submodule U of M such that M/U is finitely generated, there exists a nonzero direct summand V of M such that $$V \subseteq U$$ V ⊆ U and M/V is finitely generated. It is shown that every non-finitely generated free module is $$\mathfrak {s}$$ s -coseparable but a finitely generated free module is not, in general, $$\mathfrak {s}$$ s -coseparable. We prove that the class of $$\mathfrak {s}$$ s -coseparable modules over a right noetherian ring is closed under finite direct sums. We show that the class of commutative rings R for which every cyclic R-module is $$\mathfrak {s}$$ s -coseparable is exactly that of von Neumann regular rings. Some examples of modules M for which every direct summand of M is $$\mathfrak {s}$$ s -coseparable are provided.

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“…Case 2: Assume that M is not finitely generated. Note that M is coseparable by [8,Proposition 3.16]. Hence, M is s-coseparable by Theorem 2.7.…”

Section: Proposition 227 Letmentioning

confidence: 92%

“…Let N be a cofinite submodule of M. Then clearly N is not finitely generated. Note that N is coseparable by [8,Proposition 2.15]. Therefore, N is s-coseparable by Theorem 2.7.…”

Section: Corollary 217 Let R Be a Right Noetherian Ring And Let M Be An S-coseparable R-module Then Every Cofinite Submodule Of M Is Alsomentioning

confidence: 94%

“…Proof Note that M is coseparable. Then N is coseparable by [8,Proposition 2.12]. Since N is not finitely generated, it follows that N is s-coseparable by Theorem 2.7.…”

Section: Proposition 213mentioning

confidence: 97%

“…The following characterization provides a rich source of examples of s-coseparable modules (see [8]).…”

Section: Example 26mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On strongly coseparable modules

Ech-chaouy

Idelhadj

Tribak³

2021

Arab. J. Math.

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Simple-separable modules

Ech-chaouy,

Trıbak

2024

International Electronic Journal of Algebra

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A module $M$ over a ring is called simple-separable if every simple submodule of $M$ is contained in a finitely generated direct summand of $M$. While a direct sum of any family of simple-separable modules is shown to be always simple-separable, we prove that a direct summand of a simple-separable module does not inherit the property, in general. It is also shown that an injective module $M$ over a right noetherian ring is simple-separable if and only if $M=M_1 \oplus M_2$ such that $M_1$ is separable and $M_2$ has zero socle. The structure of simple-separable abelian groups is completely described.

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On coseparable and 𝔪-coseparable modules

Cited by 2 publications

References 13 publications

On strongly coseparable modules

On strongly coseparable modules

Simple-separable modules

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