On <i>X</i>-sub-linearly independent modules

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

doi:10.1088/1742-6596/893/1/012008

Cited by 8 publications

(7 citation statements)

References 4 publications

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“…The 𝑋-sub-exact sequence is a generalization of the exact sequence [6]. 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].…”

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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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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“…The exact sequence is a special case of X-sub-exact sequence [7]. 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].…”

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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“…Motivated by the generalization of the exact sequence to U-exact sequence and V-coexact sequence, a sub exact sequence is introduced (Fitriani et al, 2016). Furthermore, a sub-exact sequence is used to establish the X-sub-linearly independent module as a generalization of linearly independent module relative to a family of R-modules (Fitriani et al, 2017).…”

Section: Introductionmentioning

confidence: 99%

The Rank of U_V-Generated Modules

Fitriani¹,

Wijayanti²,

Surodjo³

et al. 2021

JMR

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Let U be a nonempty set of R-modules and V be a submodule of &oplus;_ʌU_λ, U_λ∈U for all λ∈Λ. A U_V generated module is a generalization of U-generated module by using the concept of V-coexact sequence. We say that an R-module N is generated by U_V if there is an epimorphism from V to N. In this paper, we introduce the definition of rank of U_V-generated modules. Furthermore, we investigate some properties of rank of UV-generated modules.

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On X-sub-linearly independent modules

Cited by 8 publications

References 4 publications

The Properties of Rough v-Coexact Sequence in Rough Group

The Properties of Rough v-Coexact Sequence in Rough Group

Category of Submodules of a Uniserial Module

The Rank of U_V-Generated Modules

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