Lifting Modules

doi:10.1007/3-7643-7573-6

Cited by 7 publications

(4 citation statements)

References 178 publications

(379 reference statements)

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“…The ring of all R-endomorphisms of M is denoted by S = End R (M ). We use the notation N ¿ M to indicate that N is small in M (i.e., for all L < A module M is said to be supplemented if every submodule of M has a supplement [4].…”

Section: Introductionmentioning

confidence: 99%

H-supplemented modules with respect to images of fully invariant submodules

Hamzekolaee

Amouzegar

2021

Proyecciones (Antofagasta)

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Lifting modules plays important roles in module theory. H-supplemented modules are a nice generalization of lifting modules which have been studied extensively recently. In this article, we introduce a proper generalization of H-supplemented modules via images of fully invariant submodules. Let F be a fully invariant submodule of a right Rmodule M. We say that M is IF -H-supplemented in case for every endomorphism φ of M, there is a direct summand D of M such that φ(F) + X = M if and only if D + X = M, for every submodule X of M. It is proved that M is IF -H-supplemented if and only if F is a dual Rickart direct summand of M for a fully invariant noncosingular submodule F of M. It is shown that the direct sum of IF –H supplemented modules is not in general IF -H-supplemented. Some sufficient conditions such that the direct sum of IF -H-supplemented modules is IF -H-supplemented are given

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Section: Introductionmentioning

confidence: 99%

H-supplemented modules with respect to images of fully invariant submodules

Hamzekolaee

Amouzegar

2021

Proyecciones (Antofagasta)

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“…For an indexed set (M α ) α∈A of modules and class of modules U, the direct sum of the traces T r(U, M) is contained in ⊕ A M α . The trace of M in an R-module N is the sum of all M-generated submodules of N (Clark et al, 2006).…”

Section: Introductionmentioning

confidence: 99%

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

Fitriani¹,

Wijayanti²,

Surodjo³

2018

JMR

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Let $\mathcal{U}$ be a non-empty set of $R$-modules. $R$-module $N$ is generated by $\mathcal{U}$ if there is an epimorphism from $\oplus_{\Lambda}U_{\lambda}$ to $N$, where $U_{\lambda} \in \mathcal{U}$, for every $\lambda \in \Lambda$. $R$-module $M$ is a subgenerator for $N$ if $N$ is isomorphic to a submodule of an $M$-generated module. In this paper, we introduce a $\mathcal{U}_{V}$-generator, where $V$ be a submo\-dule of $\oplus_{\Lambda}U_{\lambda}$, as a generalization of $\mathcal{U}$-generator by using the concept of $V$-coexact sequence. We also provide a $\mathcal{U}_{V}$-subgenerator motivated by the concept of $M$-subgenerator. Furthermore, we give some properties of $\mathcal{U}_{V}$-generated and $\mathcal{U}_{V}$-subgenerated modules related to category $\sigma[M]$. We also investigate the existence of pullback and pushout of a pair of morphisms of $\mathcal{U}_{V}$-subgenerated modules. We prove that the collection of $\mathcal{U}_{V}$-subgenerated modules is closed under submodules and factor modules.

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“…More information about (amply) supplemented lattices are in [1,2] and [3]. More results about (amply) supplemented modules are in [4,6] and [7]. The definitions of (amply) generalized supplemented modules and some properties of them are in [5].…”

Section: Introductionmentioning

confidence: 99%

Generalized supplemented lattices

Bicer¹,

Nebiyev²,

Pancar³

2018

MMN

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In this work, we define (amply) generalized supplemented lattices and investigate some properties of these lattices. In this paper, all lattices are complete modular lattices with the smallest element 0 and the greatest element 1. Let L be a lattice, 1 D a 1 _ a 2 _ : : : _ a n and the quotient sublattices a 1 =0, a 2 =0,.. . , a n =0 be generalized supplemented, then L is generalized supplemented. If L is an amply generalized supplemented lattice, then for every a 2 L, the quotient sublattice 1=a is amply generalized supplemented.

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Lifting Modules

Cited by 7 publications

References 178 publications

H-supplemented modules with respect to images of fully invariant submodules

H-supplemented modules with respect to images of fully invariant submodules

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

Generalized supplemented lattices

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