Generalization of $\mathcal{U}$-Generator and $M$-Subgenerator Related to Category $\sigma[M]$

Fitriani, Fitriani; Wijayanti, Indah Emilia; Surodjo, Budi

doi:10.5539/jmr.v10n4p101

Cited by 9 publications

(9 citation statements)

References 4 publications

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“…First, they establish the notion of an X-sub-linearly independent module. Then, by using the V-coexact sequence, Fitriani et al introduce a U V -generated module (Fitriani et al, 2018a). The U V -generated module is a dual of X-sub-linearly independent.…”

Section: Introductionmentioning

confidence: 99%

A Generalization of Projective Module

Fitriani¹,

Wijayanti²,

Faisol³

2023

JMR

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Let $V$ be a submodule of a direct sum of some elements in $\mathcal{U}$, and $X$ be a submodule of a direct sum of some elements in $\mathcal{N}$, where $\mathcal{U}$ and $\mathcal{N}$ are families of $R$-modules. A $\mathcal{U}$-free module is a generalization of a free module. According to the definition of $\mathcal{U}$-free module, we define three kinds of projective$_{\mathcal{U}}$ in this research, i.e., projective$_{\underline{\mathcal{U}}}$, projective$_{\mathcal{U}}$ module, and strictly projective$_{\mathcal{U}}$ module. The notion of strictly projective$_{\mathcal{U}}$ is a generalization of the projective module. In this research, we discuss the relationship between projective modules and the three types of modules. Furthermore, we show that the properties of $\mathcal{U}$ impact the properties of the projective$_{\mathcal{U}}$ module so that we can determine some properties of the projective$_{\mathcal{U}}$ module based on the properties of the family of $\mathcal{U}$ of $R$-modules.

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Section: Introductionmentioning

confidence: 99%

A Generalization of Projective Module

Fitriani¹,

Wijayanti²,

Faisol³

2023

JMR

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“…The concept of an exact sequence is used to define an 𝑋-sublinearly independent set [7]. In 2018, the 𝑈-generator concept was introduced based on 𝑉-coexact sequences [8]. The concept of a 𝑈 𝑣 -generator and an 𝑋-sublinear independent module family were utilized to develop by (𝑋, 𝑉)-basis and 𝑈-free modules in the same year [9].…”

Section: Introductionmentioning

confidence: 99%

The Properties of Rough v-Coexact Sequence in Rough Group

Hafifullah

Fitriani²,

Faisol³

2022

BAREKENG: J. Math. & App.

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“…As an application of a sub-exact sequence, Fitriani et al introduce an X-sub-linearly independent module [8]. Then, by using the concept of coexact sequence, Fitriani et al establish a U V -generated module [10]. This concept is a generalization of the U-generated module [13].…”

Section: Introductionmentioning

confidence: 99%

Category of Submodules of a Uniserial Module

Fitriani¹,

Wijayanti²,

Surodjo³

et al. 2021

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Let R be a ring, K, M be R-modules, L a uniserial R-module, and X a submodule of L. The triple (K, L, M ) is said to be X-sub-exact at L if the sequence K → X → M is exact. Let σ(K, L, M ) is a set of all submodules Y of L such that (K, L, M ) is Y -sub-exact. The sub-exact sequence is a generalization of an exact sequence. We collect all triple (K, L, M ) such that (K, L, M ) is an X-sub exact sequence, where X is a maximal element of σ(K, L, M ). In a uniserial module, all submodules can be compared under inclusion. So, we can find the maximal element of σ(K, L, M ). In this paper, we prove that the set σ(K, L, M ) form a category, and we denoted it by C L . Furthermore, we prove that C Y is a full subcategory of C L , for every submodule Y of L. Next, we show that if L is a uniserial module, then C L is a pre-additive category. Every morphism in C L has kernel under some conditions. Since a module factor of L is not a submodule of L, every morphism in a category C L does not have a cokernel. So, C L is not an abelian category. Moreover, we investigate a monic X-sub-exact and an epic X-sub-exact sequence. We prove that the triple (K, L, M ) is a monic X-sub-exact if and only if the triple Z-modules (Hom R (N, K), Hom R (N, L), Hom R (N, M )) is a monic Hom R (N, X)-sub-exact sequence, for all R-modules N . Furthermore, the triple (K, L, M ) is an epic X-sub-exact if and only if the triple Z-modules (Hom R (M, N ), Hom R (L, N ), Hom R (K, N )) is a monic Hom R (X, N )-sub-exact, for all R-module N .

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Generalization of $\mathcal{U}$-Generator and $M$-Subgenerator Related to Category $\sigma[M]$

Cited by 9 publications

References 4 publications

A Generalization of Projective Module

A Generalization of Projective Module

The Properties of Rough v-Coexact Sequence in Rough Group

Category of Submodules of a Uniserial Module

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