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On τ-Extending Modules
Seçil Çeken,Mustafa Alkan +1 more
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In this article, a generalization of extending (CS) modules by using the concept of τ-large submodule was introduced, and the authors give some properties of this class of modules and study their relationship with τ-closed, τ-complement submodules and the other generalisation of extending modules (τ-complemented and τ-CS, s−τ-CS modules).Abstract:
Motivated by [2] and [6], we introduce a generalization of extending (CS) modules by using the concept of τ-large submodule which was defined in [9]. We give some properties of this class of modules and study their relationship with the familiar concepts of τ-closed, τ-complement submodules and the other generalization of extending modules (τ-complemented, τ-CS, s−τ-CS modules). We are also interested in determining when a τ-divisible module is τ-extending. For a τ-extending module M with C3, we obtain a decomposition theorem that there is a submodule K of M such that \(M = \tau (M)\,\oplus\,K\) and K is τ (M)-injective. We also treat when a direct sum of τ-extending modules is τ-extending.read more
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On modules whose coclosed submodules are isomorphic to direct summands
Laba Kr Das,Mittal K. Patel +1 more
TL;DR: In this article , the authors have proposed and investigated the behaviours of virtually lifting module as the dual of virtually extending module, and proved that a finite direct sum of principally h-lifting modules are principally lifting if each component is relatively projective.
Small PPQ-injective Modules
S. Wongwai,Oam Sthityanak +1 more
TL;DR: A right R − module N is called small pseudo principally M − injective if, every R − monomorphism from a small and principal submodule of N to M can be extended to an R − homomorphism between M to N as mentioned in this paper.
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Simple continuous modules
TL;DR: In this article, it was shown that a ring is a right $V$-ring if and only if every cyclic right $R$-module is a singular simple-direct-injective module.
Left Co-Köthe and Generalized Co-Köthe Rings and Solving an Old Problem of Köthe on Non-Commutative Rings
TL;DR: In this article , the classical Köthe's problem concerning the structure of non-commutative rings with the property that every left module is a direct sum of cyclic modules was solved.
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Two criteria for locally noetherian Grothendieck categories
TL;DR: In this article , the authors give two criteria for a Grothendieck category to be locally noetherian, in terms of direct sum decompositions into uniform components and the existence of finitely injective objects that cannot be expressed as a direct sum of injectives.
References
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Book
Lectures on modules and rings
TL;DR: In this article, Baer's Test for Injectivity is used to evaluate the suitability of a set of Injective Modules over a group of Invertible Fractional Ideals.
Book
An introduction to noncommutative noetherian rings
Ken R. Goodearl,R.B Warfield +1 more
TL;DR: The 2004 introduction to noncommutative noetherian rings is intended to be accessible to anyone with a basic background in abstract algebra as mentioned in this paper, and it can be used as a second-year graduate text, or as a self-contained reference.
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A generalization of CS-modules
TL;DR: In this paper, the authors investigate two generalizations of CS-modules, i.e., if every submodule is essential in a direct sum-mand of a module, and if the module M is a CS-module or satifies (C1).
Journal ArticleDOI
Spectral gabriel topologies and relative singular functors
TL;DR: In this article, Spectral gabriel topologies and relative singular functors are studied in the context of algebraic topologies, and the authors propose a topology based on Spectral Gabbriel topology.